Solved: Section 2.7 Instantaneous AccelerationA particle

Problem 81CP A sprinter can accelerate with constant acceleration for 4.0 s before reaching top speed. He can run the 100-meter dash in 10.0 s. What is his speed as he crosses the finish Une?

Problem 80CP Careful measurements have been made of Olympic sprinters in the 100-meter dash. A quite realistic model is that the sprinter’s velocity is given by vx = a (1? e ? ht ) where t is in s, vx is in m/s, and the constants a and b are characteristic of the sprinter. Sprinter Carl Lewis’s run at the 1987 World Championships is modeled with a = 11.81 m/s and b = 0.6887 s?1. a. What was Lewis’s acceleration at t = 0 s, 2.00 s, and 4.00 s? ________________ b. Find an expression for the distance traveled at lime t. ________________ c. Your expression from part b is a transcendental equation, meaning that you can’t solve it for t.However, it’s not hard to use trial and error to find the time needed to travel a specific distance. To the nearest 0.01 s, find the time Lewis needed to sprint 100.0 m. His official time was 0.01 s more than your answer, showing that this model is very good, but not perfect.

Problem 1E Alan leaves Los Angeles at 8:00 am to drive to San Francisco, 400 mi away. He travels at a steady 50 mph. Beth leaves Los Angeles at 9:00 am and drives a steady 60 mph. a. Who gets to San Francisco first? b. How long does the first to arrive have to wait for the second?

Problem 82CP A rubber ball is shot straight up from the ground with speed v 0. Simultaneously, a second rubber ball at height h directly above the first ball is dropped from rest. a. At what height above the ground do the balls collide? Your answer will be an algebraic expression in terms of h, v 0 and g. ________________ b. What is the maximum value of h for which a collision occurs before the first ball falls back to the ground? ________________ c. For what value of h does the collision occur at the instant when the first ball is at its highest point?

Problem 83CP The Starship Enterprise returns from warp drive to ordinary space with a forward speed of 50 km/s. To the crew’s great surprise, a Klingon ship is 100 km directly ahead, traveling in the same direction at a mere 20 km/s. Without evasive action, the Enterprise will overtake and collide with the Klingons in just slightly over 3.0 s. The Enterprise’s computers react instantly to brake the ship. What magnitude acceleration does the Enterprise need to just barely avoid a collision with the Klingon ship? Assume the acceleration is constant. Hint: Draw a position-versus-time graph showing the motions of both the Enterprise and the Klingon ship. Let x 0 = 0 km be the location of the Enterprise as it returns from warp drive. How do you show graphically the situation in which the collision is “barely avoided”? Once you decide what it looks like graphically, express that situation mathematically.

For Questions 1 through 3, interpret the position graph given in each figure by writing a very short “story” of what is happening. Be creative! Have characters and situations! Simply saying that “a car moves 100 meters to the right” doesn’t qualify as a story. Your stories should make specific reference to information you obtain from the graph, such as distance moved or time elapsed.

For Questions 1 through 3, interpret the position graph given in each figure by writing a very short “story” of what is happening. Be creative! Have characters and situations! Simply saying that “a car moves 100 meters to the right” doesn’t qualify as a story. Your stories should make specific reference to information you obtain from the graph, such as distance moved or time elapsed.

Figure Q2.5 shows a position-versus-time graph for the motion of objects A and B as they move along the same axis. a. At the instant \(t=1 \mathrm{~s}\), is the speed of A greater than, less than, or equal to the speed of B ? Explain. b. Do objects A and B ever have the same speed? If so, at what time or times? Explain. Equation Transcription: Text Transcription: t=1 s

Problem 3E Section 2.1 Uniform Motion Julie drives 100 mi to Grandmother’s house. On the way to Grandmother’s, Julie drives half the distance at 40 mph and half the distance at 60 mph. On her return trip, she drives half the time at 40 mph and half the time at 60 mph. a. What is Julie’s average speed on the way to Grandmother’s house? ________________ b. What is her average speed on the return trip?

Section 2.1 Uniform Motion Larry leaves home at 9:05 and runs at constant speed to the lamppost seen in Figure Ex2.2. He reaches the lamppost at 9:07, immediately turns, and runs to the tree. Larry arrives at the tree at 9:10. a. What is Larry’s average velocity, in m/min, during each of these two intervals. b. What is Larry’s average velocity for the entire run?

For Questions 1 through 3, interpret the position graph given in each figure by writing a very short “story” of what is happening. Be creative! Have characters and situations! Simply saying that “a car moves 100 meters to the right” doesn’t qualify as a story. Your stories should make specific reference to information you obtain from the graph, such as distance moved or time elapsed.

Figure Q2.4 shows a position-versus-time graph for the motion of objects A and B as they move along the same axis. a. At the instant \(t=1 \mathrm{~s}\), is the speed of A greater than, less than, or equal to the speed of B? Explain. b. Do objects A and B ever have the same speed? If so, at what time or times? Explain. Equation Transcription: Text Transcription: t=1 s

Section 2.3 Finding Position from Velocity A particle starts from \(x_{0}=10 \mathrm{~m}\) at \(t_{0}=0 \mathrm{~s}\) and moves with the velocity graph shown in Figure EX2.6. a. Does this particle have a turning point? If so, at what time? b. What is the object's position at \(t=2 \mathrm{~s}, 3 \mathrm{~s}\), and \(4 \mathrm{~s}\)? Equation Transcription: Text Transcription: x_0=10 m t_0=0 t=2 s, 3 s, 4 s

Section 2.3 Finding Position from Velocity Figure Ex 2.7 is a somewhat idealized graph of the velocity of blood in the ascending aorta during one beat of the heart. Approximately how far, in cm, does the blood move during one beat?

Section 2.1 Uniform Motion Figure Ex2.4 is the position-versus-time graph of a jogger. What is the jogger's velocity at \(t=10\) \(\mathrm{s}\), at \(t=25 \mathrm{~s}\), and at \(t=35 \mathrm{~s}\)? Equation Transcription: Text Transcription: t=0 s t=25 s t=35 s

Section 2.3 Finding Position from Velocity FIGURE EX2.5 shows the position graph of a particle. a. Draw the particle's velocity graph for the interval \(0 \mathrm{~s} \leq t \leq 4 \mathrm{~s}\). b. Does this particle have a turning point or points? If so, at what time or times? Equation Transcription: Text Transcription: 0 s leq t leq 4 s

Figure Q2.7 shows the position-versus-time graph for a moving object. At which lettered point or points: a. Is the object moving the fastest? b. Is the object moving to the left? c. Is the object speeding up? d. Is the object turning around?

Figure Q2.6 shows the position-versus-time graph for a moving object. At which lettered point or points: a. Is the object moving the slowest? b. Is the object moving the fastest? c. Is the object at rest? d. Is the object moving to the left?

Figure Q2.8 shows six frames from the motion diagrams of two moving cars, A and B. a. Do the two cars ever have the same position at one instant of time? If so, in which frame number (or numbers)? b. Do the two cars ever have the same velocity at one instant of time? If so, between which two frames?

Problem 11CQ a. Give an example of a vertical motion with a positive velocity and a negative acceleration. b. Give an example of a vertical motion with a negative velocity and a negative acceleration.

Section 2.4 Motion with Constant Acceleration Figure Ex2.7 showed the velocity graph of blood in the aorta. Estimate the blood’s acceleration during each phase of the motion, speeding up and slowing down.

Section 2.4 Motion with Constant Acceleration Figure EX2.9 shows the velocity graph of a particle. Draw the particle’s acceleration graph for the interval \(0 \mathrm{~s} \leq t \leq 4 \mathrm{~s}\). Give both axes an appropriate numerical scale. Equation Transcription: Text Transcription: 0 s leq t leq 4 s

Problem 9CQ You’re driving along the highway at a steady speed of 60 mph when another driver decides to pass you. At the moment when the front of his car is exactly even with the front of your car, and you turn your head to smile at him, do the two cars have equal velocities? Explain.

Section 2.3 Finding Position from Velocity Figure EX 2.8 shows the velocity graph for a particle having initial position \(x_{0}=0 \mathrm{~m}\) at \(t_{0}=0 \mathrm{~s}\). a. At what time or times is the particle found at \(x=35 \mathrm{~m}\)? Work with the geometry of the graph, not with kinematic equations. b. Draw a motion diagram for the particle. Equation Transcription: Text Transcription: x_0=0 m at t_0=0 s x=35 m

Problem 10CQ A bicycle is traveling east. Can its acceleration vector ever point west? Explain.

Section 2.4 Motion with Constant Acceleration Figure EX2.11 shows the velocity graph of a particle moving along the \(x\)-axis. Its initial position is \(x_{0}=2.0 \mathrm{~m}\) at \(t_{0}=0 \mathrm{~s}\). At \(t=2.0 \mathrm{~s}\), what are the particle's (a) position, (b) velocity, and (c) acceleration? Equation Transcription: Text Transcription: x_0=2.0 m t_0=0 s t=2.0 s

Problem 13CQ A rock is thrown (not dropped) straight down from a bridge into the river below. a. Immediately after being released, is the magnitude of the rock’s acceleration greater than g , less than g , or equal to g ? Explain. b. Immediately before hitting the water, is the magnitude of the the rock’s acceleration greater than g , less than g , or equal to g ? Explain.

Problem 14CQ A rubber ball dropped from a height of 2 m bounces back to a height of 1 m. Draw the ball’s position, velocity, and acceleration graphs, stacked vertically, from the instant you release it until it returns to its maximum bounce height. Pay close attention to the time the ball is in contact with the ground; this is a short interval of time, but it’s not zero.

Problem 12CQ A ball is thrown straight up into the air. At each of the following instants, is the ball’s acceleration ay equal to g , -g, 0, < g, or >g? a. Just after leaving your hand? b. At the very top (maximum height)? c. Just before hitting the ground?

Section 2.4 Motion with Constant Acceleration Figure EX2.12 shows the velocity-versus-time graph for a particle moving along the \(x\)-axis. Its initial position is \(x_{0}=2.0 \mathrm{~m}\) at \(t_{0}=0 \mathrm{~s}\). a. What are the particle's position, velocity, and acceleration at \(t=1.0 \mathrm{~s}\)? b. What are the particle's position, velocity, and acceleration at \(t=3.0 \mathrm{~s}\) ? Equation Transcription: Text Transcription: x-axis x_0=2.0 m t_0=0 s t=1.0 s t=3.0 s

Problem 13E Section 2.4 Motion with Constant Acceleration A jet plane is cruising at 300 m/s when suddenly the pilot turns the engines up to full throttle. After traveling 4.0 km, the jet is moving with a speed of 400 m/s. What is the jet’s acceleration, assuming it to be a constant acceleration?

Problem 14E Section 2.4 Motion with Constant Acceleration When you sneeze, the air in your lungs accelerates from rest to 150 km/h in approximately 0.50 s. What is the acceleration of the air in m/s2 ?

Problem 17E Ball bearings can be made by letting spherical drops of molten metal fall inside a tall tower—called a shot tower—and solidify as they fall. a. If a bearing needs 4.0 s to solidify enough for impact, how high must the tower be? b. What is the bearing’s impact velocity?

Problem 15E Section 2.4 Motion with Constant Acceleration A speed skater moving across frictionless ice at 8.0 m/s hits a 5.0-m-wide patch of rough ice. She slows steadily, then continues on at 6.0 m/s. What is her acceleration on the rough ice?

Problem 16E Section 2.4 Motion with Constant Acceleration A Porsche challenges a Honda to a 400 m race. Because the Porsche’s acceleration of 3.5 m/s2is larger than the Honda’s 3.0 m/s2, the Honda gets a 1.0 s head start. Who wins?

Problem 18E Section 2.5 Free Fall A ball is thrown vertically upward with a speed of 19.6 m/s. a. What is the ball’s velocity and its height after 1.0, 2.0, 3.0, and 4.0 s? ________________ b. Draw the ball’s velocity-versus-time graph. Give both axes an appropriate numerical scale.

Problem 19E A student standing on the ground throws a ball straight up. The ball leaves the student’s hand with a speed of 15 m/s when the hand is 2.0 m above the ground. How long is the ball in the air before it hits the ground? (The student moves her hand out of the way.)

Problem 20E A rock is tossed straight up with a speed of 20 m/s. When it returns, it falls into a hole 10 m deep. a. What is the rock’s velocity as it hits the bottom of the hole? b. How long is the rock in the air, from the instant it is released until it hits the bottom of the hole?

Problem 22E Section 2.6 Motion on an Inclined Plane A car traveling at 30 m/s runs out of gas while traveling up a 10° slope. How far up the hill will it coast before stalling to roll back down?

Problem 21E Section 2.6 Motion on an Inclined Plane A skier is gliding along at 3.0 m/s on horizontal, frictionless snow. He suddenly starts down a 10° incline. His speed at the bottom is 15 m/s. a. What is the length of the incline? ________________ b. How long does it take him to reach the bottom?

Problem 23E Section 2.7 Instantaneous Acceleration A particle moving along the x -axis has its position described by the function x = (2t 2 ? t + 1 ) m, where t s in s. At t ? 2 s what are the particle’s (a) position, (b) velocity, and (c) acceleration?

Section 2.7 Instantaneous Acceleration Figure EX 2.25 shows the acceleration-versus-time graph of a particle moving along the \(x\)-axis. Its initial velocity is \(v_{0 x}=8.0 \mathrm{~m} / \mathrm{s}\) at \(t_{0}=0 \mathrm{~s}\). What is the particle's velocity at \(t=4.0 \mathrm{~s}\) ? Equation Transcription: Text Transcription: x-axis v_0x=8.0 m/s t_0=0 s t=4.0 s

Problem 26P A particle’s position on the x-axis is given by the function x = (t 2 ? 4r + 2) m, where t is in s. a. Make a position-versus-time graph for the interval 0 s ? t ? 5 s. Do this by calculating and plotting x every 0.5 s from 0 s to 5 s, then drawing a smooth curve through the points. ________________ b. Determine the particle’s velocity at t = 1.0 s by drawing the tangent line on your graph and measuring its slope. ________________ c. Determine the particle’s velocity at t = 1.0 s by evaluating the derivative at that instant. Compare this to your result from part b. ________________ d. Are there any turning points in the particle’s motion? If so, at what position or positions? ________________ e. Where is the particle when v x = 4.0 m/s? ________________ f. Draw a motion diagram for the particle.

Problem 24E Section 2.7 Instantaneous Acceleration A particle moving along the x -axis has its velocity described by the function vx = 2t2 m/s, where t is in s. Its initial position is x0 = 1 m at t0 = 0 s. At t = 1 s what are the particle’s (a) position, (b) velocity, and (c) acceleration?

Three particles move along the \(x\)-axis, each starting with \(v_{0 x}=10 \mathrm{~m} / \mathrm{s}\) at \(t_{0}=0 \mathrm{~s}\). In Figure P 2.27, the graph for A is a position-versus-time graph; the graph for B is a velocity-versus-time graph; the graph for C is an acceleration-versus-time graph. Find each particle's velocity at \(t= 7.0 \mathrm{~s}\). Work with the geometry of the graphs, not with kinematic equations. Equation Transcription: Text Transcription: x-axis v_0x=10 m/s t_0=0 s t=7.0 s

Figure P2.28 shows the acceleration graph for a particle that starts from rest at \(t=0 \ s\). Determine the object’s velocity at times \(t=0 \ s\), \(2 \ s\), \(4 \ s\), \(6 \ s\), and \(8 \ s\). Equation Transcription: Text Transcription: t=0 s t=0 s 2 s 4 s 6 s 8 s

Problem 30P A particle’s velocity is described by the function vx = t 2 ? 7t + 10 m/s, where t is in s. a. At what times does the particle reach its turning points? ________________ b. What is the particle’s acceleration at each of the turning points?

A block is suspended from a spring, pulled down, and released. The block’s position-versus-time graph is shown in Figure P2.29. a. At what times is the velocity zero? At what times is the velocity most positive? Most negative? b. Draw a reasonable velocity-versus-time graph.

Problem 32P An object starts from rest at x = 0 m at time t = 0 s. Five seconds later, at t = 5.0 s, the object is observed to be at x = 40 m and to have velocity vx = 11 m/s. a. Was the object’s acceleration uniform or nonuniform? Explain your reasoning. ________________ b. Sketch the velocity-versus-time graph implied by these data. Is the graph a straight line or curved? If curved, is it concave upward or downward?

Problem 31P The position of a particle is given by the function x = (2t 3 ? 9t 2 + 12) m, where t is in s. a. At what time or times is vx = 0 m/s? ________________ b. What are the particle’s position and its acceleration at this lime(s)?

Problem 33P A particle’s velocity is described by the function vx = kt 2 m/s, where k is a constant and t is in s. The particle’s position at t 0 = 0 s is x 0 = ?9.0 m. Al t 1 = 3.0 s, the particle is at x 1 = 9.0 m. Determine the value of the constant k. Be sure to include the proper units.

Draw position, velocity, and acceleration graphs for the ball shown in Figure P2.36. See Problem 35 for more information.

Problem 34P A particle’s acceleration is described by the function ax = (10 ? 1) m/s2, where t is in s. Its initial conditions are x 0 = 0 m and v 0x = 0 m/s at t = 0 s. a. At what time is the velocity again zero? ________________ b. What is the particle’s position at that time?

Draw position, velocity, and acceleration graphs for the ball shown in Figure P2.37. See Problem 35 for more information. The ball changes direction but not speed as it bounces from the reflecting wall.

Figure P2.38 shows a set of kinematic graphs for a ball rolling on a track. All segments of the track are straight lines, but some may be tilted. Draw a picture of the track and also indicate the ball’s initial condition.

A ball rolls along the frictionless track shown in Figure P2.35. Each segment of the track is straight, and the ball passes smoothly from one segment to the next without changing speed or leaving the track. Draw three vertically stacked graphs showing position, velocity, and acceleration versus time. Each graph should have the same time axis, and the proportions of the graph should be qualitatively correct. Assume that the ball has enough speed to reach the top.

Figure P2.39 shows a set of kinematic graphs for a ball rolling on a track. All segments of the track are straight lines, but some may be tilted. Draw a picture of the track and also indicate the ball’s initial condition.

Problem 40P The takeoff speed for an Airbus A320 jetliner is 80 m/s. Velocity data measured during takeoff are as shown. a. What is the takeoff speed in miles per hour? ________________ b. Is the jetliner’s acceleration constant during takeoff? Explain. ________________ c. At what time do the wheels leave the ground? ________________ d. For safety reasons, in case of an aborted takeoff, the runway must be three times the takeoff distance. Can an A320 take off safely on a 2.5-mi-long runway? (s) vx (m/s) 0 0 10 23 20 46 30 69

Problem 41P a. What constant acceleration, in SI units, must a car have to go from zero to 60 mph in 10 s? ________________ b. What fraction of g is this? ________________ c. How far has the car traveled when it reaches 60 mph? Give your answer both in SI units and in feet.

Problem 43P You are driving to the grocery store at 20 m/s. You are 110 m from an intersection when the traffic light turns red. Assume that your reaction time is 0.50 s and that your car brakes with constant acceleration. a. How far are you from the intersection when you begin to apply the brakes? ________________ b. What acceleration will bring you to rest right at the intersection? ________________ c. How long does it take you to stop after the light turns red?

Problem 42P a. How many days will it take a spaceship to accelerate to the speed of light (3.0 × 108 m/s) with the acceleration g? ________________ b. How far will it travel during this interval? ________________ c. What fraction of a light year is your answer to part b? A light year is the distance light travels in one year. NOTE We know, from Einstein’s theory of relativity, that no object can travel at the speed of light. So this problem, while interesting and instructive, is not realistic.

Problem 44P a. Suppose you are driving at speed v 0 when a sudden obstacle in the road forces you to make a quick stop. If your reaction time before applying the brakes is t R, what constant deceleration (absolute value of ax ) do you need to stop in distance d ? Assume that d is larger than the car travels during your reaction time. ________________ b. Suppose you are driving at 21 m/s when you suddenly see an obstacle 50 m ahead. If your reaction time is 0.50 s and if your car’s maximum deceleration is 6.0 m/s2, can you stop in time to avoid a collision?

Problem 54P A car accelerates at 2.0 m/s2 along a straight road. It passes two marks that are 30 m apart at times t = 4.0 s and t = 5.0 s. What was the car’s velocity at t = 0 s?

Problem 56P Ann and Carol are driving their cars along the same straight road. Carol is located at x = 2.4 mi at t = 0 h and drives at a steady 36 mph. Ann, who is traveling in the same direction, is located at x= 0.0 mi at t = 0.50 h and drives at a steady 50 mph. a. At what time does Ann overtake Carol? ________________ b. What is their position at this instant? ________________ c. Draw a position-versus-time graph showing the motion of both Ann and Carol.

Problem 55P Santa loses his footing and slides down a frictionless, snowy roof that is tilted at an angle of 30°. If Santa slides 10 m before reaching the edge, what is his speed as he leaves the roof?

Problem 57P a. A very slippery block of ice slides down a smooth ramp tilted at angle ? . The ice is released from rest at vertical height h above the bottom of the ramp. Find an expression for the speed of the ice at the bottom. ________________ b. Evaluate your answer to part a for ice released at a height of 30 cm on ramps tilted at 20° and 40°.

A toy train is pushed forward and released at \(x_{0}=2.0 \mathrm{~m}\) with a speed of \(2.0 \mathrm{~m} / \mathrm{s}\). It rolls at a steady speed for \(2.0 \mathrm{~s}\), then one wheel begins to stick. The train comes to a stop \(6.0 \mathrm{~m}\) from the point at which it was released. What is the magnitude of the train's acceleration after its wheel begins to stick? Equation Transcription: Text Transcription: x_0=2.0 m 2.0 m/s 2.0 s 6.0 m

Problem 59P Bob is driving the getaway car after the big bank robbery. He’s going 50 m/s when his headlights suddenly reveal a nail strip that the cops have placed across the road 150 m in front of him. If Bob can stop in time, he can throw the car into reverse and escape. But if he crosses the nail strip, all his tires will go flat and he will be caught. Bob’s reaction time before he can hit the brakes is 0.60 s, and his car’s maximum deceleration is 10 m/s2. Is Bob in jail?

Problem 60P One game at the amusement park has you push a puck up a long, frictionless ramp. You win a stuffed animal if the puck, at its highest point, comes to within 10 cm of the end of the ramp without going off. You give the puck a push, releasing it with a speed of 5.0 m/s when it is 8.5 m from the end of the ramp. The puck’s speed after traveling 3.0 m is 4.0 m/s. Are you a winner?

Problem 61P a. Your goal in laboratory is to launch a ball of mass m straight up so that it reaches exactly height h above the top of the launching lube. You and your lab partners will earn fewer points if the ball goes too high or too low. The launch tube uses compressed air to accelerate the ball over a distance d, and you have a table of data telling you how to set the air compressor to achieve a desired acceleration. Find an expression for the acceleration that will earn you maximum points. ________________ b. Evaluate your answer to part a to achieve a height of 3.2 m using a 45-cm-long launch tube.

Problem 62P Nicole throws a ball straight up. Chad watches the ball from a window 5.0 m above the point where Nicole released it. The ball passes Chad on the way up, and it has a speed of 10 m/s as it passes him on the way back down. How fast did Nicole throw the ball?

Problem 63P A motorist is driving at 20 m/s when she sees that a traffic light 200 m ahead has just turned red. She knows that this light stays red for 15 s, and she wants to reach the light just as it turns green again. It takes her 1.0 s to step on the brakes and begin slowing. What is her speed as she reaches the light at the instant it turas green?

When a 1984 Alfa Romeo Spider sports car accelerates at the maximum possible rate, its motion during the first \(20 \mathrm{~s}\) is extremely well modeled by the simple equation \(v_{x}^{2}=\frac{2 P}{m} t\) where \(P=3.6 \times 10^{4}\) watts is the car's power output, \(m=1200 \mathrm{~kg}\) is its mass, and \(v_{x}\) is in \(m / s\). That is, the square of the car's velocity increases linearly with time. a. What is the car's speed at \(t=10 \mathrm{~s}\) and at \(t=20 \mathrm{~s}\)? b. Find an algebraic expression in terms of \(P, m\), and \(t\), for the car's acceleration at time \(t\) c. Evaluate the acceleration at \(t=1 \mathrm{~s}$ and \(t=10 \mathrm{~s}\). d. This simple model fails for \(t\) less than about \(0.5 \mathrm{~s}\). Explain how you can recognize the failure. Equation Transcription: Text Transcription: 20 s v_x^2=2P/m t P=3.6 x 10^4watts m=1200 kg v_x m/s t=1 s t=10 s t 0.5 s

Problem 65P David is driving a steady 30 m/s when he passes Tina, who is sitting in her car at rest. Tina begins to accelerate at a steady 2.0 m/s2 at the instant when David passes. a. How far does Tina drive before passing David? ________________ b. What is her speed as she passes him?

Problem 66P A cat is sleeping on the floor in the middle of a 3.0-m-wide room when a barking dog enters with a speed of 1.50 m/s. As the dog enters, the cat (as only cats can do) immediately accelerates at 0.85 m/s2 toward an open window on the opposite side of the room. The dog (all bark and no bite) is a bit startled by the cat and begins to slow down at 0.10 m/s2 as soon as it enters the room. Does the dog catch the cat before the cat is able to leap through the window?

Problem 67P Jill has just gotten out of her car in the grocery store parking lot. The parking lot is on a hill and is tilted 3°. Twenty meters downhill from Jill, a little old lady lets go of a fully loaded shopping cart. The cart, with frictionless wheels, starts to roll straight downhill. Jill immediately starts to sprint after the cart with her top acceleration of 2.0 m/s2. How far has the cart rolled before Jill catches it?

Problem 68P As a science project, you drop a watermelon off the top of the Empire State Building, 320 m above the sidewalk. It so happens that Superman flies by at the instant you release the watermelon. Superman is headed straight down with a speed of 35 m/s. How fast is the watermelon going when it passes Superman?

Problem 69P I was driving along at 20 m/s, trying to change a CD and not watching where I was going. When 1 looked up, I found myself 45 m from a railroad crossing. And wouldn’t you know it, a train moving at 30 m/s was only 60 m from the crossing. In a split second, 1 realized that the train was going to beat me to the crossing and that I didn’t have enough distance to stop. My only hope was to accelerate enough to cross the tracks before the train arrived. If my reaction time before starting to accelerate was 0.50 s, whal minimum acceleration did my car need for me to be here today writing these words?

Problem 70P As an astronaut visiting Planet X, you’re assigned to measure the free-fall acceleration. Getting out your meter stick and stop watch, you time the fall of a heavy ball from several heights. Your data are as follows: Height (m) Fall time (s) 0.0 0.00 1.0 0.54 2.0 0.72 3.0 0.91 4.0 1.01 5.0 1.17 Analyze these data to determine the free-fall acceleration on Planet X. Your analysis method should involve fitting a straight line to an appropriate graph, similar to the analysis in Example 2.15.

Problem 71P Your engineering firm has been asked to determine the deceleration of a car during hard braking. To do so, you decide to measure the lengths of the skid marks when slopping from various initial speeds. Your data are as follows: Speed (m/s ) Skid length (m) 10 7 15 14 20 27 25 37 30 58 a. Do the data support an assertion that the deceleration is constant, independent of speed? Explain. ________________ b. Determine an experimental value for the car’s deceleration— that is, the absolute value of the acceleration. Your analysis method should involve fitting a straight line to an appropriate graph, similar- to the analysis in Example 2.15.

In Problems 72 through 75, you are given the kinematic equation or equations that are used to solve a problem. For each of these, you are to: a. Write a realistic problem for which this is the correct equation(s). Be sure that the answer your problem requests is consistent with the equation(s) given. b. Draw the pictorial representation for your problem. c. Finish the solution of the problem. \(64 \mathrm{~m}=0 \mathrm{~m}+(32 \mathrm{~m} / \mathrm{s})(4 \mathrm{~s}-0 \mathrm{~s})+\frac{1}{2} a_{x}(4 \mathrm{~s}-0 \mathrm{~s})^{2}\) Equation Transcription: Text Transcription: 64 m=0 m+(32 m/s)(4s-0 s)+1/2a_x(4s-0 s)^2

In Problems 72 through 75, you are given the kinematic equation or equations that are used to solve a problem. For each of these, you are to: a. Write a realistic problem for which this is the correct equation(s). Be sure that the answer your problem requests is consistent with the equation(s) given. b. Draw the pictorial representation for your problem. c. Finish the solution of the problem. \((10 \mathrm{~m} / \mathrm{s})^{2}=v_{0 y}^{2}-2\left(9.8 \mathrm{~m} / \mathrm{s}^{2}\right)(10 \mathrm{~m}-0 \mathrm{~m})\) Equation Transcription: Text Transcription: (10 m/s^2)v_0y ^2-2(9.8 m/s^2)(10 m-0 m)

Problem 74P In Problem, you are given the kinematic equation or equations that are used to solve a problem. For each of these, you are to: a. Write a realistic problem for which this is the correct equa-tion(s). Be sure that the answer your problem requests is consistent with the equation(s) given. ________________ b. Draw the pictorial representation for your problem. ________________ c. Finish the solution of the problem. (0 m/s)2 = (5 m/s)2 ? 2(9.8 m/s2 )(sin 10°)(x 1, ? 0 m)

\(v_{1 x}=0 \mathrm{~m} / \mathrm{s}+\left(20 \mathrm{~m} / \mathrm{s}^{2}\right)(5 s-0 s)\) \(x_{1}=0 m+(0 \mathrm{~m} / \mathrm{s})(5 \mathrm{~s}-0 \mathrm{~s})+\frac{1}{2}\left(20 \mathrm{~m} / \mathrm{s}^{2}\right)(5 \mathrm{~s}-0 s)^{2}\) \(x_{2}=x_{1}+v_{12}(10 s-5 s)\) Equation Transcription: Text Transcription: v_1x = 0 m/s + (20 m/s^2)(5 s - 0 s) x_1 = 0 m + (0 m/s)(5 s - 0 s)+ 1/2 (20 m/s^2)(5 s - 0 s)^2 x_2 = x_1 + v_12 (10 s - 5 s)

The two masses in FIGURE CP2.76 slide on frictionless wires. They are connected by a pivoting rigid rod of length \(L\). Prove that \(v_{2 x}=-v_{1 y} \tan \theta\). Equation Transcription: Text Transcription: L v_2x=v_1y tan theta

Problem 77CP A rocket is launched straight up with constant acceleration. Four seconds after liftoff, a bolt falls off the side of the rocket. The bolt hits the ground 6.0 s later. What was the rocket’s acceleration?

You're driving down the highway late one night at \(20 \mathrm{~m} / \mathrm{s}\) when a deer steps onto the road \(35 \mathrm{~m}\) in front of you. Your reaction time before stepping on the brakes is \(0.50 \mathrm{~s}\), and the maximum deceleration of your car is \(10 \mathrm{~m} / \mathrm{s}^{2}\). a. How much distance is between you and the deer when you come to a stop? b. What is the maximum speed you could have and still not hit the deer? Equation Transcription: Text Transcription: 20 m/s 35 m 0.50 s 10 m/s^2

Problem 46P The minimum stopping distance for a car traveling at a speed of 30 m/s is 60 m, including the distance traveled during the driver’s reaction time of 0.50 s. a. What is the minimum stopping distance for the same car traveling at a speed of 40 m/s? ________________ b. Draw a position-versus-time graph for the motion of the car in part a. Assume the car is at x 0= 0 m when the driver first sees the emergency situation ahead that calls for a rapid halt.

When jumping, a flea accelerates at an astounding \(1000 \mathrm{~m} / \mathrm{s}^{2}\), but over only the very short distance of \(0.50 \mathrm{~mm}\). If a flea jumps straight up, and if air resistance is neglected (a rather poor approximation in this situation), how high does the flea go? Equation Transcription: Text Transcription: 1000 m/s^2 0.50 mm

Problem 78CP Your school science club has devised a special event for homecoming. You’ve attached a rocket to the rear of a small car that has been decorated in the blue-and-gold school colors. The rocket provides a constant acceleration for 9.0 s. As the rocket shuts off, a parachute opens and slows the car at a rate of 5.0 m/s2. The car passes the judges’ box in the center of the grandstand, 990 m from the starting line, exactly 12 s after you fire the rocket. What is the car’s speed as it passes the judges?

Careful measurements have been made of Olympic sprinters in the \(100 -meter\) dash. A simple but reasonably accurate model is that a sprinter accelerates at \(3.6 \mathrm{~m} / \mathrm{s}^{2}\) for \(3 \frac{1}{3} s\), then runs at constant velocity to the finish line. a. What is the race time for a sprinter who follows this model? b. A sprinter could run a faster race by accelerating faster at the beginning, thus reaching top speed sooner. If a sprinter's top speed is the same as in part a, what acceleration would he need to run the \(100 -meter\) dash in \(9.9 \mathrm{~s}\) ? c. By what percent did the sprinter need to increase his acceleration in order to decrease his time by \(1 \%\)? Equation Transcription: Text Transcription: 100 meter 3.6 m/s^2 3 1/3s 100 meter 9.9 s 1%

A cheetah spots a Thomson's gazelle, its preferred prey, and leaps into action, quickly accelerating to its top speed of \(30 \mathrm{~m} / \mathrm{s}\), the highest of any land animal. However, a cheetah can maintain this extreme speed for only \(15 \mathrm{~s}\) before having to let up. The cheetah is \(170 \mathrm{~m}\) from the gazelle as it reaches top speed, and the gazelle sees the cheetah at just this instant. With negligible reaction time, the gazelle heads directly away from the cheetah, accelerating at \(4.6\) \(\mathrm{m} / \mathrm{s}^{2}\) for \(5.0 \mathrm{~s}\), then running at constant speed. Does the gazelle escape? Equation Transcription: Text Transcription: 30 m/s 170 m 4.6 m/s^2 5.0 s

A \(200 \mathrm{~kg}\) weather rocket is loaded with \(100 \mathrm{~kg}\) of fuel and fired straight up. It accelerates upward at \(30 \mathrm{~m} / \mathrm{s}^{2}\) for \(30 \mathrm{~s}\), then runs out of fuel. Ignore any air resistance effects. a. What is the rocket's maximum altitude? b. How long is the rocket in the air before hitting the ground? c. Draw a velocity-versus-time graph for the rocket from liftoff until it hits the ground. Equation Transcription: Text Transcription: 200 kg 100 kg 30m/s^2 30 s

Problem 50P A 1000 kg weather rocket is launched straight up. The rocket motor provides a constant acceleration for 16 s, then the motor stops. The rocket altitude 20 s after launch is 5100 m. You can ignore any effects of air resistance. a. What was the rocket’s acceleration during the first 16 s? ________________ b. What is the rocket’s speed as it passes through a cloud 5100 m above the ground?

Problem 51P A lead ball is dropped into a lake from a diving board 5.0 m above the water. After entering the water, it sinks to the bottom with a constant velocity equal to the velocity with which it hit the water. The ball reaches the bottom 3.0 s after it is released. How deep is the lake?

A hotel elevator ascends \(200 \mathrm{~m}\) with a maximum speed of \(5.0 \mathrm{~m} / \mathrm{s}\). Its acceleration and deceleration both have a magnitude of \(1.0 \mathrm{~m} / \mathrm{s}^{2}\). a. How far does the elevator move while accelerating to full speed from rest? b. How long does it take to make the complete trip from bottom to top? Equation Transcription: Text Transcription: 200 m 5.0 m/s 1.0 m/s^2

Problem 53P A car starts from rest at a stop sign. It accelerates at 4.0 m/s2 for 6.0 s, coasts for 2.0 s, and then slows down at a rate of 3.0 m/s2 for the next stop sign. How far apart are the stop signs?