What are the magnitude, the algebraic sign, and the direction of the average acceleration in each interval? a)At the beginning of the interval a body is moving toward the right along the x·axis at 5 m . s-1 and at the end of the interval it is moving toward the right at 20m-s-1• b)At the beginning it is moving toward the right at 20 m . S-l, and at the end it is moving toward the right at5m·s-l. c)At the beginning it is moving toward the left at 5 m . S-I, and at the end it is moving toward the left at 20m·s-1. d) At the beginning it is moving toward the left at 20 m . S-I, and at the end it is moving toward the left at 5m ·S-I. e) At the beginning it is moving toward the right at 20 m • S-l, and at the end it is moving toward the left at 20m s-1. f) At the beginning it is moving toward the left at 20 m . s-1, and at the end it is moving toward the right at 20m g) h) i) j) Figure 3-13 3-8 Each of the following changes in velocity takes place in a 10-s interval. ·S-I. 3-9 The makers of a certain automobile advertise that it will accelerate from 15 to 50mi 'hel in 135. Compute (a) the acceleration in ft . 5-2, and (b) the distance the car travels in this time, assuming the acceleration to be constant. 3-10 An airplane taking off from a landing field has a run of 500 m. If it starts from rest, moves with constant acceleration, and makes the run in 30 s, with what velocity in m . 5-1 did it take off? 3-11 An automobile starts from rest and acquires a velocity of 40 km per hour in 15 s. a) Compute the acceleration in kilometers per hour per second, assuming it to be constant. l b) If the automobile continues to gain velocity at the same rate, how many more seconds are needed for it to acquire a velocity of 60 km . hr- ? Figure 3-12 Find the average velocity of the body in the interval from t ::::: 0 to t = 1 s, and in the interval from t = 0 to t ::::: 4 s. 3-3 The motion of a certain body along the x-axis is described by the equation x :::; (10 em . S-2)t2. Compute the instantaneous velocity of the body at time t :::; 3 s. Let ~t first equal 0.1 s, theu 0.01 s, and finally 0.001 s. What limiting value do the results seem to be approaching? 3-4 An automobile is provided with a speedometer calibrated to read m . S-l rather than mi . hr-1. The following series of speedometer readings was obtained during a start. Time (s) o 2 4 6 8 10 12 14 16 Velocity (m . S-I) 0 0 2 5 10 15 20 22 22 a) Compute the average acceleration during each 2-s interval. Is the acceleration constant? Is it constant during any part of the trial? b) Make a velocity-time graph of the data above, using scales of 1 em = 1 s horizontally, and 1 cm :::; 2 m . 5-1 vertically. Draw a smooth curve through the plotted points. \¥hat distance is represented by 1 em2? What is the displacement in the first 8 s? \\That is the acceleration when t :::; 8 s? \\Then t :::; 13 s? \\1ben t :::; 15 s? 3-5 The graph in Fig. 3-11 shows the velocity of a body plotted as a function of time. a) Find the instantaneous acceleration at t = 3 s, at t = 7 s, and at t :::; 11 s. b) How far does the body go in the first 5 s? The first 9 s? The first 13 s? -, V,ID'S 50 45 40 35 30 25 20 15 10 5 o 24681012 Figure 3-11 3-6 Figure 3-12 is a graph of the acceleration of a body moving on the x-axis. Sketch the graphs of its velocity and coordinate as functions of time, if x = v ::::: 0 when t = O. -2 a,m";:; 3-7 Figure 3-13 is a graph of the coordinate of a body moving on the x-axis. Sketch the graphs of its velocity and acceleration as functions of time. x Parabola k) Parabola t,' 40 ~31d the distances covered by the automobile in parts a) and (b). :z A body moving with constant acceleration covers -E &t.ance between two points 60 m apart in 6 s. Its -~-;ciry as it passes the second point is 15 m . S-l. _ "';\hat is the acceleration? 0- .. >That is its velocity at the first point? -_3 A ball is released from rest and rolls down an in.:0 plane, requiring 4 s to cover a distance of 100 cm . .:.. '";\hat is its acceleration in cm . S-2? (b) How many '2:::::meters would it have fallen vertically in the same -'-",? =-4 The "reaction time" of the average automobile ~~, tl is about 0.7 s. (The reaction time is the interval =-=-=--een the perception of a signal to stop and the applica --= of brakes.) If an automobile can accelerate at -5:::l . S-2, compute the total distance covered in coming :;;;: a StOp after a signal is observed: (a) from an initial ~ity of 15 m . s-1, (b) from an initial velocity of .3:=. S-1. _5 At the instant the traffic lights turn green, an auto=oJe that has been waiting at an intersection starts "'-""'2d with a constant acceleration of 2 m . S-2. At the v-'e instant a truck, traveling with a constant velocity of :::'::::l . s-1, overtakes and passes the automobile. (a) How =-o~ beyond its starting point will the automobile overtake ::::'c ITuck? (b) How fast will it be traveling? 6 The engineer of a passenger train traveling at sights a freight train whose caboose is 200 m ~.ad on the same track. The freight train is traveling in ~e same direction as the passenger train with a velocity of :;:; m' S-l. The engineer of the ;=:::l . S-l passenger train immediately 2.,?plies the brakes, causing a constant acceleration of -1 m . S-2, while the freight train continues with constant ~d. (a) Will there be a collision? (b) If so, where will it e place? 3-17 A sled starts from rest at the top of a hill and slides down with a constant acceleration. The sled is 140 ft from the tqP of the hill 2 s after passing a point which is 92 ft :rom ~e top. Four seconds after passing the 92-ft point it \ is 198 ft from the top, and 6 s after passing the point it is 266 ft from the top. a) What is the average velocity of the sled during each of the 2-s intervals after passing the 92-ft point? b) What is the acceleration of the sled? c) What was the velocity of the sled when it passed the 92-ft point? d) How long did it take to go from the top to the 92-ft point? e) How far did the sled go during the first second after passing the 92-ft point? f) How long does it take the sled to go from the 92-ft point to the midpoint between the 92-ft and the 140-ft mark? PROBLEMS 51 g) What is the velocity of the sled as it passes the mid- point in part (f)? 3-18 A subway train starts from rest at a station and accelerates at a rate of 2 m . S-2 for 10 s. It then runs at constant speed for 30 s, and slows down at -4 m . S-2 until it stops at the next station. Find the total distance covered. 3-19 A body starts from rest, moves in a straight line with constant acceleration, and covers a distance of 64 m in 4 s. a) What is the final velocity? b) How much time was required to cover half the total distance? c) What is the distance covered in one-half the total time? d) What is the velocity when half the total distance has been covered? e) What is the velocity after one-half the total time? 3-20 The speed of an automobile going north is reduced from 30 to 20 m . S-l in a distance of 125 m. Find (a) the magnitude and direction of the acceleration, assuming it to be constant, (b) the elapsed time, and (c) the distance in which the car can be brought to rest from 20 m . s-1, assuming the acceleration of part (a). 3-21 An automobile and a truck start from rest at the same instant, with the automobile initially at some distance behind the truck. The truck has a constant acceleration of 2 m . S-2 and' the automobile an acceleration of 3 m . S-2. The automobile overtakes the truck after the truck has moved 75 m. a) How long does it take the auto to overtake the truck? b) How far was the auto behind the truck initially? c) What is the velocity of each when they are abreast? 3-22 a) With what velocity must a ball be thrown vertically upward in order to rise to a height of 20 m? b) How long will it be in the air? 3-23 A ball is thrown vertically downward from the top of a building, leaving the thrower's hand with a velocity of 10 m . S-l. a) What will be its velocity after falling for 2 s? b) How far will it fall in 2 s? c) What will be its velocity after falling 10 m? d) If it moved a distance of 1 m while in the thrower's hand, find its acceleration while in his hand. e) If the ball was released at a point 40 m above the ground, in how many seconds will it strike the ground? f) What will the velocity of the ball be when it strikes the ground? 3-24 A balloon, rising vertically with a velocity of 5 m . s-1, releases a sandbag at an instant when the balloon is 20 m above the ground. 52 MOTION ON A STRAIGHT LINE 3-24 (continued) a) Compute the position and velocity of the sandbag at the following times after its release: 1s, !s, 1 s, 2 s. b) How many seconds after its release will the bag strike the ground? c) With what velocity will it strike? 3-25 A stone is dropped from the top of a tall cliff, and 1 s later a second stone is thrown vertically downward with a velocity of 20 m . S-l. How far below the top of the cliff will the second stone overtake the first? 3-26 A ball dropped from the cornice of a building takes 0.25 s to pass a window 3 m high. How far is the top of the window below the cornice? 3-27 A ball is thrown nearly vertically upward from a point near the cornice of a tall building. It just misses the cornice on the way down, and passes a point 160 ft below its starting point 5 s after it leaves the thrower's hand. a) What was the initial velocity of the ball? b) How high did it rise above its starting point? c) What were the magnitude and direction of its velocity at the highest point? d) What were the magnitude and direction of its acceleration at the highest point? e) What was the magnitude of its velocity as it passed a point 64 ft below the starting point? 3-28 A juggler performs in a room whose ceiling is 3 m above the level of his hands. He throws a ball vertically upward so that it just reaches the ceiling. a)With what initial velocity does he till'ow the ball? b) What time is required for the ball to reach the ceiling? He till'ows a second ball upward with the same initial velocity, at the instant that the first ball is at the ceiling. c) How long after the second ball is till' own do the two balls pass each other? d) When the balls pass each other, how far are they above the juggler's hands? 3-29 An object is thrown vertically upward. It has a speed of 10 m . S-l when it has reached one-half its maximum height. a) How high does it rise? b) What are its velocity and acceleration 1 s after it is till'own? c) 3 s after? d) What is the average velocity during the first half second? 3-30 A student determined to test the law of gravity for himself walks off a skyscraper 300 m high, stopwatch in hand, and starts his free fall (zero initial velocity). Five seconds later, Superman arrives at the scene and dives off the roof to save the student. a) What must Superman's initial velocity be in order that he catch the student just before the ground is reached? b)What must be the height of the skyscraper so that even Superman can't save him? (Assume that Superman's acceleration is that of any free falling body.) c) d) 3-31 A ball is thrown vertically upward from the ground and a student gazing out of the window sees it moving upward past him at 5 m . S-l. The window is 10 m above the ground. e) How high does the ball go above the ground? f) How long does it take to go from a height of 10 m to its highest point? g) Find its velocity and acceleration! s after it left the ground. 3-32 A ball is till'own vertically upward from the ground with a velocity of 30 m . S-l. a) How long will it take to rise to its highest point? b) How high does the ball rise? c) How long after projection will the ball have a velocity of 10 m . S-l upward? d) Of 10 m . S-l downward? e) When is the displacement of the ball zero? f) When is the magnitude of the ball's velocity equal to half its velocity of projection? g) When is the magnitude of the ball's displacement equal to half the greatest height to which it rises? h) What are the magnitude and direction of the acceleration while the ball is moving upward? i)While moving downward? j) When at the highest point? 3-33 A ball rolling on an inclined plane moves with a constant acceleration. One ball is released from rest at the top of an inclined plane 18 m long and reaches the bottom 3 s later. At the same instant that the first ball is released, a second ball is projected upward along the plane from its bottom with a certain initia1'velocity. The second ball is to travel part way up the plane, stop, and return to the bottom so that it arrives simultaneously with the first ball. a) Find the acceleration. b)What must be the initial velocity of the second ball? c) How far up the plane will it travel? 3-34 The rocket-driven sled Sonic Wind No.2, used for investigating the physiological effects of large accelerations, runs on a straight, level track 3500 ft long. Starting from rest, it can reach a speed of 1000 mi . hr-1 in 1.8 s. a) Compute the acceleration, assuming it to be constant. b) What is the ratio of this acceleration to that of a free falling body, g? c) What is the distance covered? 1 d) A magazine article states that at the end of a certain run the speed of the sled was decreased from 632 mi . hr - to zero in 1.4 s, and that during this time its passenger was subjected to more than 40 times the pull of gravity (that is, the acceleration was greater than 40 g). Are these figures consistent? 1 3-35 The first stage of a rocket to launch an earth satellite will, if fired vertically upward, attain a speed of 4000 mi, he at a height of 26 mi above the earth's surface, at which point its fuel supply will be exhausted. a) Assuming constant acceleration, find the time to reach a height of 26 mi. b) How much higher would the rocket rise if it continued to "coast" vertically upward? 3-36 Suppose the acceleration of gravity were only 1.0 m . S-2, instead of 10 m . S-2. a) Estimate the height to which you could jump vertically from a standing start. b) How high could you throw a baseball? c) Estimate the maximum height of a window from which you would care to jump to a concrete sidewalk below. (Each story of an average building is about 4 m high.) d) With what speed, in miles per hour, would you strike the sidewalk? e) How many seconds would be required? 3-37 A hypothetical spaceship takes a straight-line path from the earth to the moon, a distance of about 400,000 km. Suppose it accelerates at 10 m . S-2 for the first 10 min of th~ trip, then travels at constant speed until the PROBLEMS 53 last 10 min, when it accelerates at - 10 m . S-2, just coming to rest as it reaches the moon. a) What is the maximum speed attained? b) What total time is required for the trip? c) What fraction of the total distance is traveled at constant speed? 3-38 A "moving sidewalk" in an airport terminal building moves 1 m . S-1 and is 150 m long. If a man steps on at one end and walks 2 m . S-1 relative to the moving sidewalk, how much time does he requu:e to reach the opposite end if he walks (a) in the same directio n the sidewalk is moving; (b) in the opposite direction? 3-39 Two piers A and B are located on a river, one mile apart. Two men must make round trips from pier A to pier B and return. One man is to row 1 a boat at a velocity of 4 mi, hr- relative to the water, and the other man is to walk on the shore at a velocity of 4 mi ·ru:-1. The velocity of the 1 river is 2 mi . hr- in the direction from A to B. How long does it take each man to make the round trip? 1 3-40 The driver of a car wishes to pass a truck that is traveling at a constant speed of 20 m . S-1(about 50 mi . he ). The car's maximum acceleration at this speed is 0.5 m . S-2. Initially the vehicles are separated by 25 m, and the car pulls back into the truck's lane after it is 25 m ahead of the truck. The car is 5 m long and the truck 20 m. a) How much time is required for the car to pass the truck? b) What distance does the car travel during this time? c) What is the final speed of the car, assuming its acceleration while passing the truck is constant?