Motion along a Straight Line

When a high-speed passenger train travelling at 161 km/h rounds a bend, the engineer is shocked to see that a locomotive has improperly entered onto the track from a siding and is a distance  D=676 ahead (Fig.2-32). The locomotive is moving at 29 km/h. The engineer of the high speed train immediately applies the brakes. (a)What must be the magnitude of resulting constant deceleration if a collision is to be just avoided? (b)Assume that engineer is at  x=0 when, at t=0 , he first spots the locomotive. Sketch x(t) curves for the locomotive and high speed train for the cases in which a collision is just avoided and is not quite avoided.

When startled, an armadillo will leap upward. Suppose it rises  0.544 m in first 0.200 s , (a) what is its initial speed as it leaves the ground? (b)What is its speed at a height of 0.544 m ? (c) How much higher does it go?

(a) with what speed must a ball be thrown vertically from ground level to rise to a maximum height of? (b)How long will it be in the air? (c)Sketch graphs of y, v, a vs. t for the ball. On the first two graphs, indicate the time at which 50 m is reached.

Raindrops fall 1700 m from a cloud to the ground. (a) If they were not slowed by air resistance, how fast would the drops be moving when they struck the ground? (b) Would it be safe to walk outside during a rainstorm?

At a construction site, a pipe wrench stuck the ground with a speed of 24 m/s (a)From what height was it inadvertently dropped? (b)How long was it falling? (c)Sketch graphs of y,v and a versus t for the wrench.

Question: A hot-air balloon is ascending at the rate of  $\mathbf{12}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{s}$ and $\mathbf{80}\mathbf{m}$is  above the ground when a package is dropped over the side. (a) How long does the package take to reach the ground?  (b) With what speed does it hit the ground?

As a runaway scientific balloon ascends at 19.6 m/s, one of its instrument packages breaks free of a harness and free falls. Figure 2-34 gives the vertical velocity of the package versus time, from before it breaks free to when it reaches the ground. (a) What maximum height above the breakpoint does it rise? (b) How high is the break-free point above the ground?

A bolt is dropped from a bridge under construction, falling 90 m to the valley below the bridge. (a) In how much time does it pass through the last 20% of its fall? What is its speed? (b) When it begins the last 20% of its fall. (c) When it reaches the valley beneath the bridge?

A key falls from a bridge that is 45 m above the water. It falls directly into a model boat, moving with constant velocity that is 12m  from the point of impact when the key is released. What is the speed of the boat?

A stone is dropped into a river from a bridge 43.9m above the water. Another stone is thrown vertically down 1.00s after the first is dropped. The stones strike the water at the same time. (a) What is the initial speed of the second stone? (b) Plot velocity versus time on the graph for each stone, taking zero time as the instant the first stone is released.

A ball of moist clay falls 15.0m to the ground. It is in contact with the ground for 20.0m s before stopping. (a) What is the magnitude of average acceleration of the ball during the time it is in contact with the ground? (treat the ball as a particle). (b) Is the average acceleration up or down?

Figure 2-35 shows the speed v versus height y of a ball tossed directly upward, along a y axis. Distance d is 0.40 m . The speed at height${\mathit{y}}_{\mathbf{A}}$is${\mathit{v}}_{\mathbf{A}}$.The speed at height${\mathit{y}}_{\mathbf{B}}$is$\frac{\mathbf{1}}{\mathbf{3}}{\mathit{V}}_{\mathbf{A}}$. What is speed${\mathit{V}}_{\mathbf{A}}$?

To test the quality of a tennis ball, you drop it into the floor from a height 

of 4.00 m. It rebounds to a height of 2m. If the ball is in contact with the floor 

for 12.0 ms, (a) What is the magnitude of its average acceleration during the 

contact. (b) Is the average acceleration up or down?

An object falls a distance h from rest. If it travels 0.50h in the last 1.00 s , find 

(a) the time and (b) the height of its fall. (c) Explain the physically unacceptable 

solution of the quadratic equation in t that you obtain.

Water drips from the nozzle of a shower onto the floor  below. The drops fall at regular (equal) intervals of time, the first drop striking the floor at the instant the fourth drop begins to fall. When the first drop strikes the floor, how far below the nozzle are the (a) second and (c) third drops?

A rock is thrown vertically upward from ground level at time $\mathbf{t}\mathbf{=}\mathbf{0}$  . At  $\mathbf{t}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{\hspace{0.33em}}\mathbf{s}$  ,it passes the top of a tall tower, and 1.0 s  later, it reaches its maximum height. What is the height of the tower?

A steel ball is dropped from a building’s roof and passes a window, taking 0.125 s to fall from the top to the bottom of the window, a distance of 1.20 m. It then falls to a sidewalk and bounces back past the window, moving from bottom to top in 0.125 s. Assume that the upward flight is an exact reverse of the fall. The time the ball spends below the bottom of the window is $\mathbf{\text{2.00s}}$. How tall is the building?

Question: A basketball player grabbing a rebound jump 76.0 cm vertically. How much total time (ascent and descent) does the player spend (a) in the top $\mathbf{15}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{cm}$ of this jump and (b) in the bottom 15.0 cm ? (The player seems to hang in the air at the top.)

A ball is shot vertically upward from the surface of another planet. A plot of y vs t for the ball is shown in Fig 2-36, where y is the height of the ball above its starting point and t=0 at the instant the ball is shot. The figure’s vertical scaling is set by ${\mathbf{y}}_{\mathbf{s}}\mathbf{=}\mathbf{30}\mathbf{.}\mathbf{0}\mathbf{m}\mathbf{.}$ . What are the magnitudes of  (a) the free fall acceleration on the planet and (b) the initial velocity of the ball?

Figure 2-15a gives the acceleration of a volunteer’s head and torso during a rear-end collision. At maximum head acceleration, what is the speed of (a) the head and (b) the torso?

In a forward punch in karate, the fist begins at rest at the waist and is brought rapidly forward until the arm is fully extended. The speed $\mathbf{v}\mathbf{\left(}\mathbf{t}\mathbf{\right)}$of the fist is given in figure for someone skilled in karate. The vertical scaling is set by ${\mathbf{v}}_{\mathbf{s}}\mathbf{=}\mathbf{8}\mathbf{.}\mathbf{0}\mathbf{\hspace{0.33em}}\mathbf{m}\mathbf{/}\mathbf{s}\mathbf{.}$ . How far has the first moved at (a) time $\mathbf{t}\mathbf{=}\mathbf{50}\mathbf{ }\mathbf{ms}$  (b) when the speed of the fist is maximum?

When a soccer ball is kicked toward a player and the player deflects the ball by “heading” it, the acceleration of the head during the collision can be significant. Figure 2-38 gives the measured acceleration a(t) of soccer player’s head for a bare head and a helmeted head, starting from rest. The scaling on the vertical axis is set by  ${\mathbf{a}}_{\mathbf{s}}\mathbf{=}\mathbf{200}\mathbf{\hspace{0.33em}}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}\mathbf{.}\mathbf{At}\mathbf{ }\mathbf{time}\mathbf{ }\mathbf{t}\mathbf{=}\mathbf{7}\mathbf{.}\mathbf{0}\mathbf{\hspace{0.33em}}\mathbf{ms}$ , what is the difference in the speed acquired by the bare head and the speed acquired by the helmeted head?

A salamander of the genus Hydromantes captures prey by launching its tongue as a projectile. The skeletal part of the tongue is shot forward, unfolding the rest of the tongue, until the outer portion lands on the prey, sticking to it. Figure 2-39 shows the acceleration magnitude a versus time t for the acceleration phase of the launch in a typical situation. The indicated accelerations are ${\mathbf{a}}_{\mathbf{2}}\mathbf{=}\mathbf{400}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}\mathbf{and}\mathbf{ }{\mathbf{a}}_{\mathbf{1}}\mathbf{=}\mathbf{100}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$   . What is the outward speed of the tongue at the end of the acceleration phase?

How far does the runner whose velocity time graph is shown in Fig 2-40 travel in$\mathbf{16}\mathbf{ }\mathbf{s}$?The figure’s vertical scaling is set by ${\mathbf{V}}_{\mathbf{s}}\mathbf{=}\mathbf{8}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{s}$.

Two particles move along an x axis. The position of particle 1 is given by $\mathbf{x}\mathbf{=}\mathbf{6}{\mathbf{t}}^{\mathbf{2}}\mathbf{+}\mathbf{3}\mathbf{t}\mathbf{+}\mathbf{2}$ (in meters and seconds); the acceleration of particle 2 is given by $\mathbf{a}\mathbf{=}\mathbf{-}\mathbf{8}\mathbf{t}$ (in meters per second squared and seconds), and, at $\mathbf{t}\mathbf{=}\mathbf{0}$, its velocity is $\mathbf{20}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{s}$. When the velocities of the particles match, what is their velocity?

In an arcade video game, a spot is programmed to move across the screen according to$\mathbf{x}\mathbf{=}\mathbf{9}\mathbf{.}\mathbf{00}\mathbf{t}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{750}{\mathbf{t}}^{\mathbf{3}}$, where x is distance in centimeters measured from the left edge of the screen and  $\mathbf{t}$ is time in seconds. When the spot reaches a screen edge, at either $\mathbf{x}\mathbf{=}\mathbf{0}$or $\mathbf{x}\mathbf{=}\mathbf{15}\mathbf{.}\mathbf{0}\mathbf{cm}$, t is reset to $\mathbf{0}$ and the spot starts moving again according to $\mathbf{x}\left(t\right)$. (a) At what time after starting is the spot instantaneously at rest? (b) At what value of x does this occur? (c) What is the spot’s acceleration (including sign) when this occurs? (d)Is it moving right or left just prior to coming to rest? (e) Just after?(f) At what time $\mathbf{t}\mathbf{>}\mathbf{0}$ does it first reach an edge of the screen?

A rock is shot vertically upward from the edge of the top of a tall building. The rock reaches its maximum height above the top of the building $\mathbf{1}\mathbf{.}\mathbf{60}\mathbf{s}$ after being shot. Then, after barely missing the edge of the building as it falls downward, the rock strikes the ground $\mathbf{6}\mathbf{.}\mathbf{00}\mathbf{s}$ after it is launched. In SI units: (a) with what upward velocity is the rock shot, (b) what maximum height above the top of the building is reached by the rock, and (c) how tall is the building?

At the instant the traffic light turns green, an automobile starts with a constant acceleration aof$\mathbf{2}\mathbf{.}\mathbf{2}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$. At the same instant, a truck traveling with a constant speed of $\mathbf{9}\mathbf{.}\mathbf{5}\mathbf{m}\mathbf{/}\mathbf{s}$, overtakes and passes the automobile. (a)How far beyond the traffic signal will the automobile overtake the truck? (b)How fast will the automobile be traveling at that instant?

A pilot flies horizontally at $\mathbf{1300}\mathbf{km}\mathbf{/}\mathbf{h}$, at height $\mathbf{h}\mathbf{=}\mathbf{35}\mathbf{ }\mathbf{m}$above initially level ground. However, at time t=0 , the pilot begins to fly over ground sloping upward at angle $\mathbf{\theta }\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{3}\mathbf{°}$(Fig. 2-41). If the pilot does not change the airplane’s heading, at what time   does the plane strike the ground?

To stop a car, first you require a certain reaction time to begin braking; then the car slows at a constant rate. Suppose that the total distance moved by your car during these two phases is $\mathbf{56}\mathbf{.}\mathbf{7}$m when its initial speed is $\mathbf{80}\mathbf{.}\mathbf{5}\mathbf{ }\mathbf{km}\mathbf{/}\mathbf{h}$, and $\mathbf{24}\mathbf{.}\mathbf{4}$m when its initial speed is $\mathbf{48}\mathbf{.}\mathbf{3}\mathbf{km}\mathbf{/}\mathbf{h}$ . What are (a) your reaction time and (b) the magnitude of the acceleration?

Figure 2-42 shows part of a street where traffic flow is to be controlled to allow a platoon of cars to move smoothly along the street. Suppose that the platoon leaders have just reached intersection 2, where the green appeared when they were distance d from the intersection. They continue to travel at a certain speed ${\mathbf{v}}_{\mathbf{p}}$ (the speed limit) to reach intersection 3, where the green appears when they are distance d from it. The intersections are separated by distances ${\mathbf{D}}_{\mathbf{23}}$ and ${\mathbf{D}}_{\mathbf{12}}$ . (a) What should be the time delay of the onset of green at intersection 3 relative to that at intersection 2 to keep the platoon moving smoothly?

Suppose, instead, that the platoon had been stopped by a red light at intersection 1. When the green comes on there, the leaders require a certain time ${\mathbf{t}}_{\mathbf{r}}$ to respond to the change and an additional time to accelerate at some rate a to the cruising speed ${\mathbf{V}}_{\mathbf{p}}$. (b) If the green at intersection 2 is to appear when the leaders are distance d from that intersection, how long after the light at intersection 1

turns green should the light at intersection 2 turn green?

A hot rod can accelerate from $\mathbf{0}\mathbf{ }\mathbf{to}\mathbf{ }\mathbf{60}\mathbf{ }\mathbf{km}\mathbf{/}\mathbf{h}\mathbf{ }\mathbf{in}\mathbf{ }\mathbf{5}\mathbf{.}\mathbf{4}\mathbf{ }\mathbf{s}\mathbf{.}$(a) What is its average acceleration, in $\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$, during this time? (b) How far will it travel during the , assuming its acceleration is constant? (c) From rest, how much time would it require to go a distance of 0.25 km  if its acceleration could be maintained at the value in (a)?

A red train traveling at 72 km/h and a green train traveling at $144 \mathrm{km}/\mathrm{h}$ are headed toward each other along a straight, level track. When they are 950 m apart, each engineer sees the other’s train and applies the brakes. The brakes slow each train at the rate of $1.0\mathrm{m}/{\mathrm{s}}^2$ . Is there a collision? If so, answer yes, give the speed of the red train and the speed of the green train at impact, respectively. If not, answer no and give the separation between the trains when they stop.

At time t = 0, a rock climber accidentally allows a piton to fall freely from a high point on the rock wall to the valley below him. Then, after a short delay, his climbing partner, who is 10 higher on the wall, throws a piton downward. The positions y of the pitons versus t during the falling are given in Fig. 2-43. With what speed is the second piton thrown?

A train started from rest and moved with constant acceleration. At one time, it was travelling30 m/s , and 160 m farther on it was travelling 50 m/s. Calculate (a) the acceleration (b) the time required to travel 160 m mentioned (c) the time required to attain speed of 30 m/s (d) the distance moved from rest to the time the train had a speed of 30 m/s (e) Graph x vs t and v vs t for the train, from rest.

A particle’s acceleration along an x axis is $\mathit{a}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\mathit{t}$, with t in seconds and a in meters per second squared. At $\mathit{t}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{s}$, its velocity is $\mathbf{+}\mathbf{17}\mathbf{ }\mathit{m}\mathbf{/}\mathit{s}$. What is its velocity at $\mathit{t}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{s}$?

Figure 2-44 gives the acceleration a versus time t for a particle moving along an x axis. The ‘a’ axis scale is set by ${\mathit{a}}_{\mathbf{s}}\mathbf{=}\mathbf{12}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$. At $\mathit{t}\mathbf{=}\mathbf{‐}\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{s}$, the particle’s velocity is 7.0 m/s . What is its velocity at $\mathit{t}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{s}$?

Figure 2-45 shows a simple device for measuring your reaction time. It consists of a cardboard strip marked with a scale and two large dots. A friend holds the strip vertically, with thumb and forefinger at the dot on the right in Fig. 2-45. You then position your thumb and forefinger at the other dot (on the left in Fig. 2-45), being careful not to touch the strip. Your friend releases the strip, and you try to pinch it as soon as possible after you see it begin to fall. The mark at the place where you pinch the strip gives your reaction time. (a) How far from the lower dot should you place the 50.0 m s mark? How much higher should you place the marks for (b) 100, (c) 150 , (d) 200 , and (e) 250 ms ? (For example, should the 100 ms marker be 2 times as far from the dot as the 50ms marker? If so, give an answer of 2 times. Can you find any pattern in the answers?

A rocket driven sled running on a straight, level track is used to investigate the effects of large accelerations on humans. One such sled can attain a speed of 1600 km/h in 1.8 s , starting from rest. Find (a)the acceleration (assumed constant) in terms of g(b) the distance traveled.

A mining cart is pulled up a hill at 20 km/h and then pulled back down the hill at 35 km/h through its original level. (The time required for the cart’s reversal at the top of its climb is negligible).What is the average speed of the cart for its round trip, from its original level back to its original level?

A motorcyclist who is moving along an x axis directed toward the east has an acceleration given by $\mathit{a}\mathbf{=}\left(6.1‐1.2t\right)\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$for $\mathbf{0}\mathbf{\le }\mathbf{t}\mathbf{\le }\mathbf{6}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{s}$. At $\mathit{t}\mathbf{=}\mathbf{0}$, the velocity and position of the cyclist are $\mathbf{2}\mathbf{.}\mathbf{7}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{s}$ and $\mathbf{7}\mathbf{.}\mathbf{3}\mathbf{ }\mathbf{m}$ . (a) What is the maximum speed achieved by the cyclist? (b) What total distance does the cyclist travel between $\mathit{t}\mathbf{=}\mathbf{0}$ and $\mathbf{6}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{s}$?

When the legal speed limit for New York thruway was increased from $\mathbf{55}\mathbf{ }\mathbf{mi}\mathbf{/}\mathbf{h}$ to$\mathbf{65}\mathbf{ }\mathbf{mi}\mathbf{/}\mathbf{h}$ , how much time was saved by a motorist who drove the 700 km between the buffalo entrance and the New York city exit at the legal speed limit?

A car moving with constant acceleration covered the distance between two points $\mathbf{60}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{m}$ apart in $\mathbf{6}\mathbf{.}\mathbf{00}\mathit{s}$. Its speed as it passed the second point was $\mathbf{15}\mathbf{.}\mathbf{0}\mathit{m}\mathbf{/}\mathit{s}$. (a) What was the speed at the first point? (b) What was the magnitude of the acceleration? (c) At what prior distance from the first point was the car at rest? (d) Graph x versus t and v  versus t for the car, from rest $(t =0)$.

A certain juggler usually tosses balls vertically to a height H. To what height must they be tossed if they are to spend twice as much time in the air?

A particle starts from the origin at $\mathit{t}\mathbf{=}\mathbf{0}$ and moves along the positive $\mathit{x}\mathbf{ }\mathit{a}\mathit{x}\mathit{i}\mathit{s}$. A graph of the velocity of the particle as a function of the time is shown in Fig. 2-46; the $\mathit{v}\mathbf{-}\mathit{a}\mathit{x}\mathit{i}\mathit{s}$ scale is set by vs $\mathbf{4}\mathbf{.}\mathbf{0}\mathit{m}\mathbf{/}\mathit{s}$. (a) What is the coordinate of the particle at $\mathit{t}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\mathit{s}$? (b) What is the velocity of the particle at$\mathit{t}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\mathit{s}$? (c) What the acceleration of the particle at $\mathit{t}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\mathit{s}$? (d) What is the average velocity of the particle between $\mathit{t}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{0}\mathit{s}$ and $\mathit{t}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\mathit{s}$? (e) What is the average acceleration of the particle between $\mathit{t}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{0}\mathit{s}$ and $\mathit{t}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\mathit{s}$?

A rock is dropped from a $\mathbf{100}\mathbf{ }\mathit{m}$ high cliff. How long does it take to fall (a) the first $\mathbf{50}\mathbf{ }\mathit{m}$ and (b) the second $\mathbf{50}\mathbf{ }\mathit{m}$?

Two subway stops are separated by $\mathbf{1100}\mathit{m}$. If a subway train accelerates at $\mathbf{+}\mathbf{1}\mathbf{.}\mathbf{2}\mathit{m}\mathbf{/}{\mathit{s}}^{\mathbf{2}}$ from rest through the first half of the distance and decelerates at $\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{2}\mathit{m}\mathbf{/}{\mathit{s}}^{\mathbf{2}}$ through the second half, what are (a) its travel time and (b) its maximum speed? (c) Graph x, v and a versus t for the trip.

A stone is thrown vertically upward. On its way up it passes point A with speed v, and point B ,3.00 m ,higher than A, with speed $\frac{\mathbf{1}}{\mathbf{2}}\mathbf{v}$ Calculate (a) the speed v and (b) the maximum height reached by the stone above point B.

Question: An iceboat has a constant velocity toward the east when a sudden gust of wind causes the iceboat to have a constant acceleration toward the east for a period of $\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{s}$. A plot of x versus t is shown in Fig. 2-47, where $\mathit{t}\mathbf{=}\mathbf{0}$ is taken to be the instant the wind starts to blow and the positive is toward the east. (a) What is the acceleration of the iceboat during the $\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{s}$interval? (b) What is the velocity of the iceboat at the end of the $\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{s}$ interval? (c) If the acceleration remains constant for an additional $\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{s}$, how far does the iceboat travel during this second $\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{s}$ interval?

A ball is thrown vertically downward from the top of a $\mathbf{36}\mathbf{.}\mathbf{6}\mathbf{ }\mathbf{m}$-tall building. The ball passes the top of a window that is $\mathbf{12}\mathbf{.}\mathbf{2}\mathbf{ }\mathbf{m}$ above the ground $\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{s}$ after being thrown. What is the speed of the ball as it passes the top of the window? top speed. What must this distance be if he is to achieve a time of $\mathbf{10}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{s}$ for the race?