WHAT WE HAVE LEARNED

EXAM STUDY GUIDE

In two or three dimensions, any change in the magnitude or direction of an object's velocity corresponds to acceleration.

Projectile motion of an object can be separated into motion in the

*x*-direction, described by the equations and motion in the*y*-direction, described by*x*=*x*_{0}+*v*_{x0}*t**v*=_{x}*v*_{x0}*y*=*y*_{0}+*v*_{y0}*t*−*gt*^{2}*v*_{y}=*v*_{y0}−*gt**v*^{2}_{y}=*v*^{2}_{y0}− 2*g*(*y*−*y*_{0})

The relationship between the

*x*- and*y*-coordinates for ideal projectile motion can be described by a parabola given by the formula , where*y*_{0}is the initial vertical position,*v*_{0}is the initial speed of the projectile, and*θ*_{0}is the initial angle with respect to the horizontal at which the projectile is launched.The range

*R*of a projectile is given byThe maximum height

*H*reached by an ideal projectile is given by , where*v*_{y0}is the vertical component of the initial velocity.Projectile trajectories are not parabolas when air resistance is taken into account. In general, the trajectories of realistic projectiles do not reach the maximum predicted height, and have a significantly shorter range.

The velocity of an object with respect to a stationary laboratory reference frame can be calculated using a Galilean transformation of the velocity, , where is the velocity of the object with respect to a moving reference frame and is the constant velocity of the moving reference frame with respect to the laboratory frame.

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NEW SYMBOLS

, relative velocity of an object with respect to a laboratory frame of reference

ANSWERS TO SELF - TEST OPORTUNITIES

**3.1**Use equation 3.23 and*t*= (*x*−*x*_{0})/*v*_{x0}= (*x*−*x*_{0})/(*v*_{0}cos*θ*_{0}) to find**3.2**The time to reach the top is*v*=_{y}*v*_{y0}−*gt*_{top}= 0 ⇒*t*_{top}=*v*_{y0}/*g*=*v*_{0}sin*θ*/*g*. The total flight time is*t*_{total}= 2*t*_{top}because of the symmetry of the parabolic projectile trajectory. The range is the product of the total flight time and the horizontal velocity component:*R*=*t*_{total}*v*_{x0}= 2*t*_{top}*v*_{0}cos*θ*= 2(*v*_{0}sin*θ*/*g*)*v*_{0}cos*θ*= sin(2*θ*)/*g*.

PROBLEM-SOLVING PRACTICE

Problem-Solving Guidelines

In all problems involving moving reference frames, it is important to clearly distinguish which object has what motion in which frame and relative to what. It is convenient to use subscripts consisting of two letters, where the first letter stands for a particular object and the second letter for the object it is moving relative to. The moving walkway situation discussed at the opening of Section 3.6 is a good example of this use of subscripts.

In all problems concerning ideal projectile motion, the motion in the

*x*-direction is independent of that in the*y*-direction. To solve these, you can almost always use the seven kinematical equations (3.11 through 3.17), which describe motion with constant velocity in the horizontal direction and free-fall motion with constant acceleration in the vertical direction. In general, you should avoid cookie-cutter–style application of formulas, but in exam situations, these seven kinematic equations can be your first line of defense. Keep in mind, however, that these equations work only in situations in which the horizontal acceleration component is zero and the vertical acceleration component is constant.

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SOLVED PROBLEM3.3
Time of Flight

You may have participated in Science Olympiad during middle school or high school. In one of the events in Science Olympiad, the goal is to hit a horizontal target at a fixed distance with a golf ball launched by a trebuchet. Competing teams build their own trebuchets. Your team has constructed a trebuchet that is able to launch the golf ball with an initial speed of 17.2 m/s, according to extensive tests performed before the competition.

PROBLEM

If the target is located at the same height as the elevation from which the golf ball is released and at a horizontal distance of 22.42 m away, how long will the golf ball be in the air before it hits the target?

SOLUTION

THINK

Let's first eliminate what does not work. We cannot simply divide the distance between trebuchet and target by the initial speed, because this would imply that the initial velocity vector is in the horizontal direction. Since the projectile is in free fall in the vertical direction during its flight, it would certainly miss the target. So we have to aim the golf ball with an angle larger than zero relative to the horizontal. But at what angle do we need to aim?

If the golf ball, as stated, is released from the same height as the height of the target, then the horizontal distance between the trebuchet and the target is equal to the range. Because we also know the initial speed, we can calculate the release angle. Knowing the release angle and the initial speed lets us determine the horizontal component of the velocity vector. Since this horizontal component does not change in time, the flight time is simply given by the range divided by the horizontal component of the velocity.

SKETCH

We don't need a sketch at this point because it would simply show a parabola, as for all projectile motion. However, we do not know the initial angle yet, so we will need a sketch later.

RESEARCH

The range of a projectile is given by equation 3.25:

If we know the value of this range and the initial speed, we can find the angle:

Once we have the value for the angle, we can use it to calculate the horizontal component of the initial velocity:

Finally, as noted previously, we obtain the flight time as the ratio of the range and the horizontal component of the velocity:

SIMPLIFY

If we solve the equation for the angle, sin 2*θ*_{0} =*Rg*/, we see that it has two solutions: one for an angle of less than 45° and one for an angle of more than 45°. Figure 3.20 plots the function sin 2*θ*_{0} (in red) for all possible values of the initial angle *θ*_{0} and shows where that curve crosses the plot of *gR*/*v*_{0}^{2} (blue horizontal line). We call the two solutions *θ*_{a} and *θ*_{b}.

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Algebraically, these solutions are given as

Substituting this result into the formula for the horizontal component of the velocity results in

CALCULATE

Inserting numbers, we find:

ROUND

The range was specified to four significant figures, and the initial speed to three. Therefore, we also state our final results to three significant figures:

Note that both solutions are valid in this case, and the team can select either one.

DOUBLE-CHECK

Back to the approach that does not work: simply taking the distance from the trebuchet to the target and dividing it by the speed. This incorrect procedure leads to *t*_{min} = *d*/*v*_{0} = 1.30 s. We write *t*_{min} to symbolize this value to indicate that it is some lower boundary representing the case in which the initial velocity vector points horizontally and in which we neglect the free-fall motion of the projectile. Thus, *t*_{min} serves as an absolute lower boundary, and it is reassuring to note that the shorter time we obtained above is a little larger than this lowest possible, but physically unrealistic, value.

SOLVED PROBLEM3.4
Moving Deer

The zookeeper who captured the monkey in Example 3.1 now has to capture a deer. We found that she needed to aim directly at the monkey for that earlier capture. She decides to fire directly at her target again, indicated by the bull's-eye in Figure 3.21.

PROBLEM

Where will the tranquilizer dart hit if the deer is *d* = 25 m away from the zookeeper and running from her right to her left with a speed of *v*_{d} = 3.0 m/s? The tranquilizer dart leaves her rifle horizontally with a speed of *v*_{0} = 90. m/s.

SOLUTION

THINK

The deer is moving at the same time as the dart is falling, which introduces two complications. It is easiest to think about this problem in the moving reference frame of the deer. In that frame, the sideways horizontal component of the dart's motion has a constant velocity of . The vertical component of the motion is again a free-fall motion. The total displacement of the dart is then the vector sum of the displacements caused by both of these motions.

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SKETCH

We draw the two displacements in the reference frame of the deer (Figure 3.22). The blue arrow is the displacement due to the free-fall motion, and the red arrow is the sideways horizontal motion of the dart in the reference frame of the deer. The advantage of drawing the displacements in this moving reference frame is that the bull's-eye is attached to the deer and is moving with it.

RESEARCH

First, we need to calculate the time it takes the tranquilizer dart to move 25 m in the direct line of sight from the gun to the deer. Because the dart leaves the rifle in the horizontal direction, the initial forward horizontal component of the dart's velocity vector is 90 m/s. For projectile motion, the horizontal velocity component is constant. Therefore, for the time the dart takes to cross the 25-m distance, we have

During this time, the dart falls under the influence of gravity, and this vertical displacement is

Also, during this time, the deer has a sideways horizontal displacement in the reference frame of the zookeeper of *x* = −*v*_{d}*t* (the deer moves to the left, hence the negative value of the horizontal velocity component). Therefore, the displacement of the dart in the reference frame of the deer is (see Figure 3.22)

SIMPLIFY

Substituting the expression for the time into the equations for the two displacements results in

CALCULATE

We are now ready to put in the numbers:

ROUND

Rounding our results to two significant figures gives:

The net effect is the vector sum of the sideways horizontal and vertical displacements, as indicated by the green diagonal arrow in Figure 3.22: The dart will miss the deer and hit the ground behind the deer.

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DOUBLE-CHECK

Where should the zookeeper aim? If she wants to hit the running deer, she has to aim approximately 0.38 m above and 0.83 m to the left of her intended target. A dart fired in this direction will hit the deer, but not in the center of the bull's-eye. Why? With this aim, the initial velocity vector does not point in the horizontal direction. This lengthens the flight time, as we just saw in Solved Problem 3.3. A longer flight time translates into a larger displacement in both *x*- and *y*-directions. This correction is small, but calculating it is a bit too involved to show here.

MULTIPLE-CHOICE QUESTIONS

**3.1**An arrow is shot horizontally with a speed of 20 m/s from the top of a tower 60 m high. The time to reach the ground will be8.9 s

7.1 s

3.5 s

2.6 s

1.0 s

**3.2**A projectile is launched from the top of a building with an initial velocity of 30 m/s at an angle of 60° above the horizontal. The magnitude of its velocity at*t*= 5 s after the launch is−23.0 m/s

7.3 m/s

15.0 m/s

27.5 m/s

50.4 m/s

**3.3**A ball is thrown at an angle between 0° and 90° with respect to the horizontal. Its velocity and acceleration vectors are parallel to each other at0°

45°

60°

90°

none of the above

**3.4**An outfielder throws the baseball to first base, located 80 m away from the fielder, with a velocity of 45 m/s. At what launch angle above the horizontal should he throw the ball for the first baseman to catch the ball in 2 s at the same height?50.74°

25.4°

22.7°

18.5°

12.6°

**3.5**A 50-g ball rolls off a countertop and lands 2 m from the base of the counter. A 100-g ball rolls off the same counter top with the same speed. It lands from the base of the counter.less than 1 m

1 m

2 m

4 m

more than 4 m

**3.6**For a given initial speed of an ideal projectile, there is (are) launch angle(s) for which the range of the projectile is the same.only one

two different

more than two but a finite number of

only one if the angle is 45° but otherwise two different

an infinite number of

**3.7**A cruise ship moves southward in still water at a speed of 20.0 km/h, while a passenger on the deck of the ship walks toward the east at a speed of 5.0 km/h. The passenger's velocity with respect to Earth is20.6 km/h, at an angle of 14.04° east of south.

20.6 km/h, at an angle of 14.04° south of east.

25.0 km/h, south.

25.0 km/h, east.

20.6 km/h, south.

**3.8**Two cannonballs are shot from different cannons at angles*θ*_{01}= 20° and*θ*_{02}= 30°, respectively. Assuming ideal projectile motion, the ratio of the launching speeds,*v*_{01}/*v*_{02}, for which the two cannonballs achieve the same range is0.742 m

0.862 m

1.212 m

1.093 m

2.222 m

**3.9**The acceleration due to gravity on the Moon is 1.62 m/s^{2}, approximately a sixth of the value on Earth. For a given initial velocity*v*_{0}and a given launch angle*θ*_{0}, the ratio of the range of an ideal projectile on the Moon to the range of the same projectile on Earth,*R*_{Moon}/*R*_{Earth}, will be6 m

3 m

12 m

5 m

1 m

**3.10**A baseball is launched from the bat at an angle*θ*_{0}= 30° with respect to the positive*x*-axis and with an initial speed of 40 m/s, and it is caught at the same height from which it was hit. Assuming ideal projectile motion (positive*y*-axis upward), the velocity of the ball when it is caught is(20.00 + 34.64

*ŷ*) m/s.(−20.00 + 34.64

*ŷ*) m/s.(34.64 − 20.00

*ŷ*) m/s.(34.64 + 20.00

*ŷ*) m/s.

**3.11**In ideal projectile motion, the velocity and acceleration of the projectile at its maximum height are, respectively,horizontal, vertical downward.

horizontal, zero.

zero, zero.

zero, vertical downward.

zero, horizontal.

**3.12**In ideal projectile motion, when the positive*y*-axis is chosen to be vertically upward, the*y*-component of the acceleration of the object during the ascending part of the motion and the*y*-component of the acceleration during the descending part of the motion are, respectively,positive, negative.

negative, positive.

positive, positive.

negative, negative.

**3.13**In ideal projectile motion, when the positive*y*-axis is chosen to be vertically upward, the*y*-component of the velocity of the object during the ascending part of the motion and the*y*-component of the velocity during the descending part of the motion are, respectively,positive, negative.

negative, positive.

positive, positive.

negative, negative.

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QUESTIONS

**3.14**A ball is thrown from ground at an angle between 0° and 90°. Which of the following remain constant:*x*,*y*,*v*_{x},*v*_{y},*a*_{x},*a*_{y}?**3.15**A ball is thrown straight up by a passenger in a train that is moving with a constant velocity. Where would the ball land—back in his hands, in front of him, or behind him? Does your answer change if the train is accelerating in the forward direction? If yes, how?**3.16**A rock is thrown at an angle 45° below the horizontal from the top of a building. Immediately after release will its acceleration be greater than, equal to, or less than the acceleration due to gravity?**3.17**Three balls of different masses are thrown horizontally from the same height with different initial speeds, as shown in the figure. Rank in order, from the shortest to the longest, the times the balls take to hit the ground.**3.18**To attain maximum height for the trajectory of a projectile, what angle would you choose between 0° and 90°, assuming that you can launch the projectile with the same initial speed independent of the launch angle. Explain your reasoning.**3.19**An airplane is traveling at a constant horizontal speed*v*, at an altitude*h*above a lake when a trapdoor at the bottom of the airplane opens and a package is released (falls) from the plane. The airplane continues horizontally at the same altitude and velocity. Neglect air resistance.What is the distance between the package and the plane when the package hits the surface of the lake?

What is the horizontal component of the velocity vector of the package when it hits the lake?

What is the speed of the package when it hits the lake?

**3.20**Two cannonballs are shot in sequence from a cannon, into the air, with the same muzzle velocity, at the same launch angle. Based on their trajectory and range, how can you tell which one is made of lead and which one is made of wood. If the same cannonballs where launched in vacuum, what would the answer be?**3.21**One should never jump off a moving vehicle (train, car, bus, etc.). Assuming, however, that one does perform such a jump, from a physics standpoint, what would be the best direction to jump in order to minimize the impact of the landing? Explain.**3.22**A boat travels at a speed of*v*_{BW}relative to the water in a river of width*D*. The speed at which the water is flowing is*v*_{W}.Prove that the time required to cross the river to a point exactly opposite the starting point and then to return is

Also prove that the time for the boat to travel a distance

*D*downstream and then return is*T*_{1}= 2*Dv*_{B}/ ().

**3.23**A rocket-powered hockey puck is moving on a (frictionless) horizontal air-hockey table. The*x*- and*y*-components of its velocity as a function of time are presented in the graphs below. Assuming that at*t*= 0 the puck is at (*x*_{0},*y*_{0}) = (1, 2), draw a detailed graph of the trajectory*y*(*x*).**3.24**In a three-dimensional motion, the*x*-,*y*-, and*z*-coordinates of the object as a function of time are given byDescribe the motion and the trajectory of the object in an

*xyz*coordinate system.**3.25**An object moves in the*xy*-plane. The*x*- and*y*-coordinates of the object as a function of time are given by the following equations:*x*(*t*) = 4.9*t*^{2}+ 2*t*+1 and*y*(*t*) = 3*t*+ 2. What is the velocity vector of the object as a function of time? What is its acceleration vector at a time*t*= 2 s?**3.26**A particle's motion is described by the following two parametric equations:where the displacements are in meters and

*t*is the time, in seconds.Draw a graph of the particle's trajectory (that is, a graph of

*y*versus*x*).Determine the equations that describe the

*x*- and*y*-components of the*velocity*,*v*_{x}and*v*_{y}*,*as functions of time.Draw a graph of the particle's speed as a function of time.

**3.27**In a proof-of-concept experiment for an antiballistic missile defense system, a missile is fired from the ground of a shooting range toward a stationary target on the ground. The system detects the missile by radar, analyzes in real time its parabolic motion, and determines that it was fired from a distance*x*_{0}= 5000 m, with an initial speed of 600 m/s at a launch angle*θ*_{0}= 20°. The defense system then calculates the required time delay measured from the launch of the missile and fires a small rocket situated at*y*_{0}= 500 m with an initial velocity of*v*_{0}m/s at a launch angle*α*_{0}= 60° in the*yz*-plane, to intercept the missile. Determine the initial speed*v*_{0}of the intercept rocket and the required time delay.**3.28**A projectile is launched at an angle of 45° above the horizontal. What is the ratio of its horizontal range to its maximum height? How does the answer change if the initial speed of the projectile is doubled?**3.29**In a projectile motion, the horizontal range and the maximum height attained by the projectile are equal.What is the launch angle?

If everything else stays the same, how should the launch angle,

*θ*_{0}, of a projectile be changed for the range of the projectile to be halved?

**3.30**An air-hockey puck has a model rocket rigidly attached to it. The puck is pushed from one corner along the long side of the 2.00-m long air-hockey table, with the rocket pointing along the short side of the table, and at the same time the rocket is fired. If the rocket thrust imparts an acceleration of 2.00 m/s^{2}to the puck, and the table is 1.00 m wide, with what minimum initial velocity should the puck be pushed to make it to the opposite short side of the table without bouncing off either long side of the table? Draw the trajectory of the puck for three initial velocities:*v*<*v*_{min},*v*=*v*_{min}, and*v*>*v*_{min}. Neglect friction and air resistance.**3.31**On a battlefield, a cannon fires a cannonball up a slope, from ground level, with an initial velocity*v*_{0}at an angle*θ*_{0}above the horizontal. The ground itself makes an angle*α*above the horizontal (*α*<*θ*_{0}). What is the range*R*of the cannonball, measured along the inclined ground? Compare your result with the equation for the range on horizontal ground (equation 3.25).**3.32**Two swimmers with a soft spot for physics engage in a peculiar race that models a famous optics experiment: the Michelson-Morley experiment. The race takes place in a river 50.0 m wide that is flowing at a steady rate of 3.00 m/s. Both swimmers start at the same point on one bank and swim at the same speed of 5.00 m/s*with respect to the stream*. One of the swimmers swims directly across the river to the closest point on the opposite bank and then turns around and swims back to the starting point. The other swimmer swims*along*the river bank, first upstream a distance exactly equal to the width of the river and then downstream back to the starting point. Who gets back to the starting point first?

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PROBLEMS

A blue problem number indicates a worked-out solution is available in the Student Solutions Manual. One **•** and two **••** indicate increasing level of problem difficulty.

Section 3.2

**3.33**What is the magnitude of an object's average velocity if an object moves from a point with coordinates*x*= 2.0 m,*y*= −3.0 m to a point with coordinates*x*= 5.0 m,*y*= −9.0 m in a time interval of 2.4 s?**3.34**A man in search of his dog drives first 10 mi northeast, then 12 mi straight south, and finally 8 mi in a direction 30° north of west. What are the magnitude and direction of his resultant displacement?**3.35**During a jaunt on your sailboat, you sail 2.00 km east, 4.00 km southeast, and an additional distance in an unknown direction. Your final position is 6.00 km directly east of the starting point. Find the magnitude and direction of the third leg of your journey.**3.36**A truck travels 3.02 km north and then makes a 90° left turn and drives another 4.30 km. The whole trip takes 5.00 min.With respect to a two-dimensional coordinate system on the surface of Earth such that the

*y*-axis points north, what is the net displacement vector of the truck for this trip?What is the magnitude of the average velocity for this trip?

**•3.37**A rabbit runs in a garden such that the*x*- and*y*-components of its displacement as function of times are given by*x*(*t*) = −0.45*t*^{2}− 6.5*t*+ 25 and*y*(*t*) = 0.35*t*^{2}+ 8.3*t*+ 34. (Both*x*and*y*are in meters and*t*is in seconds.)Calculate the rabbit's position (magnitude and direction) at

*t*= 10 s.Calculate the rabbit's velocity at

*t*= 10 s.Determine the acceleration vector at

*t*= 10 s.

**••****3.38**Some rental cars have a GPS unit installed, which allows the rental car company to check where you are at all times and thus also know your speed at any time. One of these rental cars is driven by an employee in the company's lot and, during the time interval from 0 to 10 s, is found to have a position vector as a function of time ofWhat is the distance of this car from the origin of the coordinate system at

*t*= 5.00 s?What is the velocity vector as a function of time?

What is the speed at

*t*= 5.00 s?

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Extra credit: Can you produce a plot of the trajectory of the car in the *xy*-plane?

Section 3.3

**3.39**A skier launches off a ski jump with a horizontal velocity of 30.0 m/s (and no vertical velocity component). What are the magnitudes of the horizontal and vertical components of her velocity the instant before she lands 2.00 s later?**3.40**An archer shoots an arrow from a height of 1.14 m above ground with an initial velocity of 47.5 m/s and an initial angle of 35.2° above the horizontal. At what time after the release of the arrow from the bow will the arrow be flying exactly horizontally?**3.41**A football is punted with an initial velocity of 27.5 m/s and an initial angle of 56.7°. What is its hang time (the time until it hits the ground again)?**3.42**You serve a tennis ball from a height of 1.8 m above the ground. The ball leaves your racket with a speed of 18.0 m/s at an angle of 7.00° above the horizontal. The horizontal distance from the court's baseline to the net is 11.83 m, and the net is 1.07 m high. Neglect spin imparted on the ball as well as air resistance effects. Does the ball clear the net? If yes, by how much? If not, by how much did it miss?**3.43**Stones are thrown horizontally with the same velocity from two buildings. One stone lands twice as far away from its building as the other stone. Determine the ratio of the heights of the two buildings.**3.44**You are practicing throwing darts in your dorm. You stand 3.0 m from the wall on which the board hangs. The dart leaves your hand with a horizontal velocity at a point 2.0 m above the ground. The dart strikes the board at a point 1.65 m from the ground. Calculate:the time of flight of the dart;

the initial speed of the dart;

the velocity of the dart when it hits the board.

**•3.45**A football player kicks a ball with a speed of 22.4 m/s at an angle of 49° above the horizontal from a distance of 39 m from the goal line.By how much does the ball clear or fall short of clearing the crossbar of the goalpost if that bar is 3.05 m high?

What is the vertical velocity of the ball at the time it reaches the goalpost?

**•****3.46**An object fired at an angle of 35.0° above the horizontal takes 1.50 s to travel the last 15.0 m of its vertical distance and the last 10.0 m of its horizontal distance. With what velocity was the object launched?**•****3.47**A conveyor belt is used to move sand from one place to another in a factory. The conveyor is tilted at an angle of 14.0° from the horizontal and the sand is moved without slipping at the rate of 7.00 m/s. The sand is collected in a big drum 3.00 m below the end of the conveyor belt. Determine the horizontal distance between the end of the conveyor belt and the middle of the collecting drum.**•****3.48**Your friend's car is parked on a cliff overlooking the ocean on an incline that makes an angle of 17.0° below the horizontal. The brakes fail, and the car rolls from rest down the incline for a distance of 29.0 m to the edge of the cliff, which is 55.0 m above the ocean, and, unfortunately, continues over the edge and lands in the ocean.Find the car's position relative to the base of the cliff when the car lands in the ocean.

Find the length of time the car is in the air.

**•3.49**An object is launched at a speed of 20.0 m/s from the top of a tall tower. The height*y*of the object as a function of the time*t*elapsed from launch is*y*(*t*) = −4.9*t*^{2}+ 19.32*t*+ 60, where*h*is in meters and*t*is in seconds. Determine:the height

*H*of the tower;the launch angle;

the horizontal distance traveled by the object before it hits the ground.

**•****3.50**A projectile is launched at a 60° angle above the horizontal on level ground. The change in its velocity between launch and just before landing is found to be . What is the initial velocity of the projectile? What is its final velocity just before landing?**••****3.51**The figure shows the paths of a tennis ball your friend drops from the window of her apartment and of the rock you throw from the ground at the same instant. The rock and the ball collide at*x*= 50.0 m,*y*= 10.0 m and*t*= 3.00 s. If the ball was dropped from a height of 54.0 m, determine the velocity of the rock initially and at the time of its collision with the ball.

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Section 3.4

**3.52**For a science fair competition, a group of high school students build a kicker-machine that can launch a golf ball from the origin with a velocity of 11.2 m/s and initial angle of 31.5° with respect to the horizontal.Where will the golf ball fall back to the ground?

How high will it be at the highest point of its trajectory?

What is the ball's velocity vector (in Cartesian components) at the highest point of its trajectory?

What is the ball's acceleration vector (in Cartesian components) at the highest point of its trajectory?

**3.53**If you want to use a catapult to throw rocks and the maximum range you need these projectiles to have is 0.67 km, what initial speed do your projectiles have to have as they leave the catapult?**3.54**What is the maximum height above ground a projectile of mass 0.79 kg, launched from ground level, can achieve if you are able to give it an initial speed of 80.3 m/s?**•****3.55**During one of the games, you were asked to punt for your football team. You kicked the ball at an angle of 35.0° with a velocity of 25.0 m/s. If your punt goes straight down the field, determine the average velocity at which the running back of the opposing team standing at 70.0 m from you must run to catch the ball at the same height as you released it. Assume that the running back starts running as the ball leaves your foot and that the air resistance is negligible.**•****3.56**By trial and error, a frog learns that it can leap a maximum horizontal distance of 1.3 m. If, in the course of an hour, the frog spends 20% of the time resting and 80% of the time performing identical jumps of that maximum length, in a straight line, what is the distance traveled by the frog?**•3.57**A circus juggler performs an act with balls that he tosses with his right hand and catches with his left hand. Each ball is launched at an angle of 75° and reaches a maximum height of 90 cm above the launching height. If it takes the juggler 0.2 s to catch a ball with his left hand, pass it to his right hand and toss it back into the air, what is the maximum number of balls he can juggle?**••****3.58**In an arcade game, a ball is launched from the corner of a smooth inclined plane. The inclined plane makes a 30.0° angle with the horizontal and has a width of*w*= 50.0 cm. The spring-loaded launcher makes an angle of 45.0° with the lower edge of the inclined plane. The goal is to get the ball into a small hole at the opposite corner of the inclined plane. With what initial velocity should you launch the ball to achieve this goal?**••****3.59**A copy-cat daredevil tries to reenact Evel Knievel's 1974 attempt to jump the Snake River Canyon in a rocket-powered motorcycle. The canyon is*L*= 400. m wide, with the opposite rims at the same height. The height of the launch ramp at one rim of the canyon is*h*= 8.00 m above the rim, and the angle of the end of the ramp is 45.0° with the horizontal.What is the minimum launch speed required for the daredevil to make it across the canyon? Neglect the air resistance and wind.

Famous after his successful first jump, but still recovering from the injuries sustained in the crash caused by a strong bounce upon landing, the daredevil decides to jump again but to add a landing ramp with a slope that will match the angle of his velocity at landing. If the height of the landing ramp at the opposite rim is 3.00 m, what is the new required launch speed, and how far from the launch rim and at what height should the edge of the landing ramp be?

Section 3.5

**3.60**A golf ball is hit with an initial angle of 35.5° with respect to the horizontal and an initial velocity of 83.3 mph. It lands a distance of 86.8 m away from where it was hit. By how much did the effects of wind resistance, spin, and so forth reduce the range of the golf ball from the ideal value?

Section 3.6

**3.61**You are walking on a moving walkway in an airport. The length of the walkway is 59.1 m. If your velocity relative to the walkway is 2.35 m/s and the walkway moves with a velocity of 1.77 m/s, how long will it take you to reach the other end of the walkway?**3.62**The captain of a boat wants to travel directly across a river that flows due east with a speed of 1.00 m/s. He starts from the south bank of the river and heads toward the north bank. The boat has a speed of 6.10 m/s with respect to the water. What direction (in degrees) should the captain steer the boat? Note that 90° is east, 180° is south, 270° is west, and 360° is north.**3.63**The captain of a boat wants to travel directly across a river that flows due east. He starts from the south bank of the river and heads toward the north bank. The boat has a speed of 5.57 m/s with respect to the water. The captain steers the boat in the direction 315°. How fast is the water flowing? Note that 90° is east, 180° is south, 270° is west, and 360° is north.**•****3.64**The air speed indicator of a plane that took off from Detroit reads 350. km/h and the compass indicates that it is heading due east to Boston. A steady wind is blowing due north at 40.0 km/h. Calculate the velocity of the plane with reference to the ground. If the pilot wishes to fly directly to Boston (due east) what must the compass read?**•3.65**You want to cross a straight section of a river that has a uniform current of 5.33 m/s and is 127. m wide. Your motorboat has an engine that can generate a speed of 17.5 m/s for your boat. Assume that you reach top speed right away (that is, neglect the time it takes to accelerate the boat to top speed).If you want to go directly across the river with a 90° angle relative to the riverbank, at what angle relative to the riverbank should you point your boat?

How long will it take to cross the river in this way?

In which direction should you aim your boat to achieve minimum crossing time?

What is the minimum time to cross the river?

What is the minimum speed of your boat that will still enable you to cross the river with a 90° angle relative to the riverbank?

**•****3.66**During a long airport layover, a physicist father and his 8-year-old daughter try a game that involves a moving walkway. They have measured the walkway to be 42.5 m long. The father has a stopwatch and times his daughter. First, the daughter walks with a constant speed in the same direction as the conveyor. It takes 15.2 s to reach the end of the walkway. Then, she turns around and walks with the same speed relative to the conveyor as before, just this time in the opposite direction. The return leg takes 70.8 s. What is the speed of the walkway conveyor relative to the terminal, and with what speed was the girl walking?**•****3.67**An airplane has an air speed of 126.2 m/s and is flying due north, but the wind blows from the northeast to the southwest at 55.5 m/s. What is the plane's actual ground speed?

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Additional Problems

**3.68**A cannon is fired from a hill 116.7 m high at an angle of 22.7° with respect to the horizontal. If the muzzle velocity is 36.1 m/s, what is the speed of a 4.35-kg cannonball when it hits the ground 116.7 m below?**3.69**A baseball is thrown with a velocity of 31.1 m/s at an angle of*θ*= 33.4° above horizontal. What is the horizontal component of the ball's velocity at the highest point of the ball's trajectory?**3.70**A rock is thrown horizontally from the top of a building with an initial speed of*v*= 10.1 m/s. If it lands*d*= 57.1 m from the base of the building, how high is the building?**3.71**A car is moving at a constant 19.3 m/s, and rain is falling at 8.9 m/s straight down. What angle*θ*(in degrees) does the rain make with respect to the horizontal as observed by the driver?**3.72**You passed the salt and pepper shakers to your friend at the other end of a table of height 0.85 m by sliding them across the table. They both missed your friend and slid off the table, with velocities of 5 m/s and 2.5 m/s, respectively.Compare the times it takes the shakers to hit the floor.

Compare the distance that each shaker travels from the edge of the table to the point it hits the floor.

**3.73**A box containing food supplies for a refugee camp was dropped from a helicopter flying horizontally at a constant elevation of 500. m. If the box hit the ground at a distance of 150. m horizontally from the point of its release, what was the speed of the helicopter? With what speed did the box hit the ground?**3.74**A car drives straight off the edge of a cliff that is 60.0 m high. The police at the scene of the accident note that the point of impact is 150. m from the base of the cliff. How fast was the car traveling when it went over the cliff?**3.75**At the end of the spring term, a high school physics class celebrates by shooting a bundle of exam papers into the town landfill with a homemade catapult. They aim for a point that is 30.0 m away and at the same height from which the catapult releases the bundle. The initial horizontal velocity component is 3.90 m/s. What is the initial velocity component in the vertical direction? What is the launch angle?**3.76**Salmon often jump upstream through waterfalls to reach their breeding grounds. One salmon came across a waterfall 1.05 m in height, which she jumped in 2.1 s at an angle of 35° to continue upstream. What was the initial speed of her jump?**3.77**A firefighter, 60 m away from a burning building, directs a stream of water from a ground-level fire hose at an angle of 37° above the horizontal. If the water leaves the hose at 40.3 m/s, which floor of the building will the stream of water strike? Each floor is 4 m high.**3.78**A projectile leaves ground level at an angle of 68° above the horizontal. As it reaches its maximum height,*H*, it has traveled a horizontal distance,*d*, in the same amount of time. What is the ratio*H*/*d*?**3.79**The McNamara Delta terminal at the Metro Detroit Airport has moving walkways for the convenience of the passengers. Robert walks beside one walkway and takes 30.0 s to cover its length. John simply stands on the walkway and covers the same distance in 13.0 s. Kathy walks on the walkway with the same speed as Robert's. How long does Kathy take to complete her stroll?**3.80**Rain is falling vertically at a constant speed of 7.00 m/s. At what angle from the vertical do the raindrops appear to be falling to the driver of a car traveling on a straight road with a speed of 60.0 km/h?**3.81**To determine the gravitational acceleration at the surface of a newly discovered planet, scientists perform a projectile motion experiment. They launch a small model rocket at an initial speed of 50.0 m/s and an angle of 30.0° above the horizontal and measure the (horizontal) range on flat ground to be 2165 m. Determine the value of*g*for the planet.**3.82**A diver jumps from a 40.0 m high cliff into the sea. Rocks stick out of the water for a horizontal distance of 7.00 m from the foot of the cliff. With what minimum horizontal speed must the diver jump off the cliff in order to clear the rocks and land safely in the sea?**3.83**An outfielder throws a baseball with an initial speed of 32 m/s at an angle of 23° to the horizontal. The ball leaves his hand from a height of 1.83 m. How long is the ball in the air before it hits the ground?**•****3.84**A rock is tossed off the top of a cliff of height 34.9 m. Its initial speed is 29.3 m/s, and the launch angle is 29.9° with respect to the horizontal. What is the speed with which the rock hits the ground at the bottom of the cliff?**•3.85**During the 2004 Olympic Games, a shot putter threw a shot put with a speed of 13.0 m/s at an angle of 43° above the horizontal. She released the shot put from a height of 2 m above the ground.How far did the shot put travel in the horizontal direction?

How long was it until the shot put hit the ground?

**•****3.86**A salesman is standing on the Golden Gate Bridge in a traffic jam. He is at a height of 71.8 m above the water below. He receives a call on his cell phone that makes him so mad that he throws his phone horizontally off the bridge with a speed of 23.7 m/s.How far does the cell phone travel horizontally before hitting the water?

What is the speed with which the phone hits the water?

**•****3.87**A security guard is chasing a burglar across a rooftop, both running at 4.2 m/s. Before the burglar reaches the edge of the roof, he has to decide whether or not to try jumping to the roof of the next building, which is 5.5 m away and 4.0 m lower. If he decides to jump horizontally to get away from the guard, can he make it? Explain your answer.**•****3.88**A blimp is ascending at the rate of 7.50 m/s at a height of 80.0 m above the ground when a package is thrown from its cockpit horizontally with a speed of 4.70 m/s.How long does it take for the package to reach the ground?

With what velocity (magnitude and direction) does it hit the ground?

**•3.89**Wild geese are known for their lack of manners. One goose is flying northward at a level altitude of*h*_{g}= 30.0 m above a north-south highway, when it sees a car ahead in the distance moving in the southbound lane and decides to deliver (drop) an “egg.” The goose is flying at a speed of*v*_{g}= 15.0 m/s, and the car is moving at a speed of*v*_{c}= 100.0 km/h.Given the details in the figure, where the separation between the goose and the front bumper of the car,

*d*= 104.1 m, is specified at the instant when the goose takes action, will the driver have to wash the windshield after this encounter? (The center of the windshield is*h*_{c}= 1.00 m off the ground.)If the delivery is completed, what is the relative velocity of the “egg” with respect to the car at the moment of the impact?

**•****3.90**You are at the mall on the top step of a down escalator when you lean over laterally to see your 1.80 m tall physics professor on the bottom step of the adjacent up escalator. Unfortunately, the ice cream you hold in your hand falls out of its cone as you lean. The two escalators have identical angles of 40.0° with the horizontal, a vertical height of 10.0 m, and move at the same speed of 0.400 m/s. Will the ice cream land on your professor's head? Explain. If it does land on his head, at what time and at what vertical height does that happen? What is the relative speed of the ice cream with respect to the head at the time of impact?**•****3.91**A basketball player practices shooting three-pointers from a distance of 7.50 m from the hoop, releasing the ball at a height of 2.00 m above ground. A standard basketball hoop's rim top is 3.05 m above the floor. The player shoots the ball at an angle of 48° with the horizontal. At what initial speed must he shoot to make the basket?**•****3.92**Wanting to invite Juliet to his party, Romeo is throwing pebbles at her window with a launch angle of 37° from the horizontal. He is standing at the edge of the rose garden 7.0 m below her window and 10.0 m from the base of the wall. What is the initial speed of the pebbles?**•3.93**An airplane flies horizontally above the flat surface of a desert at an altitude of 5.00 km and a speed of 1000. km/h. If the airplane is to drop a care package that is supposed to hit a target on the ground, where should the plane be with respect to the target when the package is released? If the target covers a circular area with a diameter of 50.0 m, what is the “window of opportunity” (or margin of error allowed) for the release time?**•****3.94**A plane diving with constant speed at an angle of 49.0° with the vertical, releases a package at an altitude of 600. m. The package hits the ground 3.50 s after release. How far horizontally does the package travel?**••****3.95**10.0 seconds after being fired, a cannonball strikes a point 500. m horizontally from and 100. m vertically above the point of launch.With what initial velocity was the cannonball launched?

What maximum height was attained by the ball?

What is the magnitude and direction of the ball's velocity just before it strikes the given point?

**••****3.96**Neglect air resistance for the following. A soccer ball is kicked from the ground into the air. When the ball is at a height of 12.5 m, its velocity is (5.6 + 4.1*ŷ*) m/s.To what maximum height will the ball rise?

What horizontal distance will be traveled by the ball?

With what velocity (magnitude and direction) will it hit the ground?

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