Mechanics

You are standing at rest at a bus stop. A bus moving at a constant speed of 5.00 m/s passes you. When the rear of the bus 12 m is  past you, you realize that it is your bus, so you start to run toward it with a constant acceleration of$\mathbf{0}\mathbf{.}\mathbf{960}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$. How far would you have to run before you catch up with the rear of the bus, and how fast must you be running then? Would an average college student be physically able to accomplish this?

The driver of a car wishes to pass a truck that is traveling at a constant speed of $\mathbf{20}\mathbf{.}\mathbf{0}\mathit{m}\mathbf{/}\mathit{s}$ (about$\mathbf{41}\mathbf{mil}\mathbf{/}\mathbf{h}$ ). Initially, the car is also traveling at$\mathbf{20}\mathbf{.}\mathbf{0}\mathit{m}\mathbf{/}\mathit{s}$ , and its front bumper is $24.0m$ behind the truck’s rear bumper. The car accelerates at a constant 0.600$m/s^2$, then pulls back into the truck’s lane when the rear of the car is $26.0m$ ahead of the front of the truck. The car is  long, and the truck is 21. 0m long. (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?

An object’s velocity is measured to be ${\mathbf{V}}_{\mathbf{x}}\left(t\right)\mathbf{=}\mathbf{a}\mathbf{-}{\mathbf{\beta t}}^{\mathbf{2}}$, where $\mathbf{a}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{00}\mathbf{m}\mathbf{/}\mathbf{s}$ and $\mathbf{\beta }\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{00}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{3}}$. At t = 0 the object is at $\mathbf{x}\mathbf{=}\mathbf{0}$ . (a) Calculate the object’s position and acceleration as functions of time. (b) What is the object’s maximum positive displacement from the origin?

The acceleration of a particle is given by ${\mathbf{a}}_x\left(\mathbf{t}\right)=-\mathbf{2}.\mathbf{00}\mathbf{m}/{\mathbf{s}}^2+(\mathbf{3}.\mathbf{00}\mathbf{m}/{\mathbf{s}}^3)\mathbf{t}$ . (a) Find the initial velocity  ${\mathbf{v}}_{0x}$ such that the particle will have the same x-coordinate at $\mathbf{t}=\mathbf{4}.\mathbf{00}\mathbf{s}$ as it had at  $\mathbf{t}\mathbf{=}\mathbf{0}$. (b) What will be the velocity at  $\mathbf{t}=\mathbf{4}.\mathbf{00}\mathbf{s}$?

You are on the roof of the physics building, $\mathbf{46}.\mathbf{0}\mathbf{m}$ above the ground (Fig. P2.70). Your physics professor, who is $\mathbf{1}.\mathbf{80}\mathbf{m}$ tall, is walking alongside the building at a constant speed of $\mathbf{1}.\mathbf{20}\mathbf{m}/\mathbf{s}$. If you wish to drop an egg on your professor’s head, where should the professor be when you release the egg? Assume that the egg is in free fall.

                                          (Fig. P2.70)

A certain volcano on earth can eject rocks vertically to a maximum height H. (a) How high (in terms of H) would these rocks go if a volcano on Mars ejected them with the same initial velocity? The acceleration due to gravity on Mars is 3.71 m/s2; ignore air resistance on both planets. (b) If the rocks are in the air for a time T on earth, for how long (in terms of T) would they be in the air on Mars?

An entertainer juggles balls while doing other activities. In one act, she throws a ball vertically upward, and while it is in the air, she runs to and from a table 5.50 m away at an average speed of 3.00 m/s, returning just in time to catch the falling ball. (a) With what minimum initial speed must she throw the ball upward to accomplish this feat? (b) How high above its initial position is the ball just as she reaches the table?

While following a treasure map, you start at an old oak tree. You first walk $825 m$ directly south, then turn and walk $\mathbf{1}\mathbf{.}\mathbf{25}\mathbf{ }\mathit{k}\mathit{m}$at $\mathbf{30}\mathbf{.}\mathbf{0}\mathbf{°}$west of north, and finally walk $\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{ }\mathit{k}\mathit{m}$at $\mathbf{32}\mathbf{.}\mathbf{0}\mathbf{°}$north of east, where you find the treasure: a biography of Isaac Newton!

 (a) To return to the old oak tree, in what direction should you head and how far will you walk? Use components to solve this problem. 

(b) To see whether your calculation in part (a) is reasonable, compare it with a graphical solution drawn roughly to scale.

Sam heaves a 16-lb shot straight up, giving it a constant upward acceleration from rest of 35.0 m/s² for 64.0 cm. He releases it 2.20 m above the ground. Ignore air resistance. (a) What is the speed of the shot when Sam releases it? (b) How high above the ground does it go? (c) How much time does he have to get out of its way before it returns to the height of the top of his head, 1.83 m above the ground?

A flowerpot falls off a windowsill and passes the window of the story below. Ignore air resistance. It takes the pot 0.380 s to pass from the top to the bottom of this window, which is 1.90 m high. How far is the top of the window below the windowsill from which the flowerpot fell?

In the first stage of a two-stage rocket, the rocket is fired from the launch pad starting from rest but with a constant acceleration of 3.50 m/s² upward. At 25.0 s after launch, the second stage fires for 10.0 s, which boosts the rocket’s velocity to 132.5 m/s upward at 35.0 s after launch? This firing uses up all of the fuel, however, so after the second stage has finished firing; the only force acting on the rocket is gravity. Ignore air resistance. (a) Find the maximum height that the stage two rockets reach above the launch pad. (b) How much time after the end of the stage-two firing will it take for the rocket to fall back to the launch pad? (c) How fast will the stage-two rocket be moving just as it reaches the launch pad?

During your summer internship for an aerospace company, you are asked to design a small research rocket. The rocket is to be launched from rest from the earth’s surface and is to reach a maximum height of 960 m above the earth’s surface. The rocket’s engines give the rocket an upward acceleration of 16.0 m/s2 during the time T that they fire. After the engines shut off, the rocket is in free fall. Ignore air resistance. What must be the value of T in order for the rocket to reach the required altitude?

A physics teacher performing an outdoor demonstration suddenly falls from rest off a high cliff and simultaneously shouts “Help.” When she has fallen for 3.0 s, she hears the echo of her shout from the valley floor below. The speed of sound is 340 m/s. (a) how tall is the cliff? (b) If we ignore air resistance, how fast will she be moving just before she hits the ground?

A helicopter carrying Dr. Evil takes off with a constant upward acceleration of $\mathbf{5}\mathbf{.}\mathbf{0}\mathit{m}\mathbf{/}{\mathit{s}}^{\mathbf{2}}$ . Secret agent Austin Powers jumps on just as the helicopter lifts off the ground. After the two men struggle for 10.0 s, Powers shuts off the engine and steps out of the helicopter. Assume that the helicopter is in free fall after its engine is shut off, and ignore the effects of air resistance. (a) What is the maximum height above ground reached by the helicopter? (b) Powers deploys a jet pack strapped on his back 7.0 s after leaving the helicopter, and then he has a constant downward acceleration with magnitude $\mathbf{2}\mathbf{.}\mathbf{0}\mathit{m}\mathbf{/}{\mathit{s}}^{\mathbf{2}}$ . How far is Powers above the ground when the helicopter crashes into the ground?

Cliff Height. You are climbing in the High Sierra when you suddenly find yourself at the edge of a fog-shrouded cliff. To find the height of this cliff, you drop a rock from the top; 8.00 s later you hear the sound of the rock hitting the ground at the foot of the cliff. (a) If you ignore air resistance, how high is the cliff if the speed of sound is 330 m/s? (b) Suppose you had ignored the time it takes the sound to reach you. In that case, would you have overestimated or underestimated the height of the cliff? Explain.

An object is moving along the x-axis. At t = 0 it has velocity ${\mathbf{v}}_{\mathbf{0}}\mathbf{x}\mathbf{=}\mathbf{20}\mathbf{.}\mathbf{0}\mathbf{m}\mathbf{/}\mathbf{s}$. Starting at time t = 0 it has acceleration ${\mathbf{a}}_{\mathbf{x}}\mathbf{=}\mathbf{-}\mathbf{Ct}$, where C has units of $\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{3}}$. (a) What is the value of C if the object stops in 8.00 s after t = 0? (b) For the value of C calculated in part (a), how far does the object travel during the 8.00 s?

A ball is thrown straight up from the ground with speed ${\mathbf{v}}_{\mathbf{0}}$. At the same instant, a second ball is dropped from rest from a height H, directly above the point where the first ball was thrown upward. There is no air resistance. (a) Find the time at which the two balls collide. (b) Find the value of H in terms of ${\mathbf{v}}_{\mathbf{0}}$ and g such that at the instant when the balls collide, the first ball is at the highest point of its motion.

Cars A and B travel in a straight line. The distance of A from the starting point is given as a function of time by ${\mathbf{x}}_{\mathbf{A}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{\alpha t}\mathbf{+}{\mathbf{\beta t}}^{\mathbf{2}}$, with $\mathbf{\alpha }\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{60}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{s}$ and $\mathbf{\beta }\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{20}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$. The distance of B from the starting point is ${\mathbf{x}}_{\mathbf{B}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{\gamma t}}^{\mathbf{2}}\mathbf{-}{\mathbf{\delta t}}^{\mathbf{3}}$, with $\mathbf{\gamma }\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{80}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$ and $\mathbf{\delta }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{20}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{3}}$ . (a) Which car is ahead just after the two cars leave the starting point? (b) At what time(s) are the cars at the same point? (c) At what time(s) is the distance from A to B neither increasing nor decreasing? (d) At what time(s) do A and B have the same acceleration?

Question: In a physics lab experiment, you release a small steel ball at various heights above the ground and measure the ball’s speed just before it strikes the ground. You plot your data on a graph that has the release height (in meters) on the vertical axis and the square of the final speed (in ) on the horizontal axis. In this graph your data points lie close to a straight line. (a) Using and ignoring the effect of air resistance, what is the numerical value of the slope of this straight line? (Include the correct units.) The presence of air resistance reduces the magnitude of the downward acceleration, and the effect of air resistance increases as the speed of the object increases. You repeat the experiment, but this time with a tennis ball as the object being dropped. Air resistance now has a noticeable effect on the data. (b) Is the final speed for a given release height higher than, lower than, or the same as when you ignored air resistance? (c) Is the graph of the release height versus the square of the final speed still a straight line? Sketch the qualitative shape of the graph when air resistance is present.

Question: A model car starts from rest and travels in a straight line. A smartphone mounted on the car has an app that transmits the magnitude of the car’s acceleration (measured by an accelerometer) every second. The results are given in the table:

Each measured value has some experimental error. (a) Plot acceleration versus time and find the equation for the straight line that gives the best fit to the data. (b) Use the equation for a (t) that you found in part (a) to calculate v(t) , the speed of the car as a function of time. Sketch the graph of versus . Is this graph a straight line? (c) Use your result from part (b) to calculate the speed of the car at t = 5.00s. (d) Calculate the distance the car travels between t = 0 and  t = 5.00s.

In the vertical jump, an athlete starts from a crouch and jumps upward as high as possible. Even the best athletes spend little more than 1.00 s in the air (their “hang time”). Treat the athlete as a particle and let${\mathit{Y}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$${\mathit{Y}}_{\mathbf{m}\mathbf{a}\mathbf{x}}$be his maximum height above the floor. To explain why he seems to hang in the air, calculate the ratio of the time he is above${\mathit{Y}}_{\mathbf{m}\mathbf{a}\mathbf{x}}\mathbf{/}\mathbf{2}$to the time it takes him to go from the floor to that height. Ignore air resistance.

A student is running at her top speed of 5.0 m/s to catch a bus, which is stopped at the bus stop. When the student is still 40.0 m from the bus, it starts to pull away, moving with a constant acceleration of 0.170 m/s². (a) For how much time and what distance does the student have to run at 5.0 m/s before she overtakes the bus? (b) When she reaches the bus, how fast is the bus traveling? (c) Sketch an x-t graph for both the student and the bus. Take x = 0 at the initial position of the student. (d) The equations you used in part (a) to find the time have a second solution, corresponding to a later time for which the student and bus are again at the same place if they continue their specified motions. Explain the significance of this second solution. How fast is the bus traveling at this point? (e) If the student’s top speed is 3.5 m/swill she catch the bus? (f) What is the minimum speed the student must have to just catch up with the bus?For what time and what distance does she have to run in that case?

A ball is thrown straight up from the edge of the roof of a building. A second ball is dropped from the roof $\mathbf{1}\mathbf{.}\mathbf{00}\mathbf{ }\mathbf{s}$ later. Ignore air resistance. (a) If the height of the building is $\mathbf{20}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{m}$, what must the initial speed of the first ball be if both are to hit the ground at the same time? On the same graph, sketch the positions of both balls as a function of time, measured from when the first ball is thrown. Consider the same situation, but now let the initial speed ${\mathbf{v}}_{\mathbf{0}}$ of the first ball be given and treat the height of the building as an unknown. (b) What must the height of the building be for both balls to reach the ground at the same time if (i) ${\mathbf{v}}_{\mathbf{0}\mathbf{ }}\mathbf{is}\mathbf{ }\mathbf{9}\mathbf{.}\mathbf{5}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{s}\mathbf{ }$? (c) If ${\mathbf{v}}_{\mathbf{0}}$is greater than some value${\mathbf{v}}_{\mathbf{max}}$, no value of exists that allows both balls to hit the ground at the same time. Solve for ${\mathbf{v}}_{\mathbf{max}}$. The value ${\mathbf{v}}_{\mathbf{max}}$ has a simple physical interpretation. What is it? (d) If ${\mathbf{v}}_{\mathbf{0}}$ is less than some value ${\mathbf{v}}_{\mathbf{min}}$, no value of exists that allows both balls to hit the ground at the same time. Solve for ${\mathbf{v}}_{\mathbf{min}}$. The value ${\mathbf{v}}_{\mathbf{min}}$also has a simple physical interpretation. What is it?

The human circulatory system is closed—that is, the blood pumped out of the left ventricle of the heart into the arteries is constrained to a series of continuous, branching vessels as it passes through the capillaries and then into the veins as it returns to the heart. The blood in each of the heart’s four chambers comes briefly to rest before it is ejected by contraction of the heart muscle.

If the contraction of the left ventricle lasts 250ms and the speed of blood flow in the aorta (the large artery leaving the heart) is 0.80 m/s at the end of the contraction, what is the average acceleration of a red blood cell as it leaves the heart? (a) 310 m/s²; (b) 31 m/s²; (c) 3.2 m/s²; (d) 0.32 m/s².

The human circulatory system is closed—that is, the blood pumped out of the left ventricle of the heart into the arteries is constrained to a series of continuous, branching vessels as it passes through the capillaries and then into the veins as it returns to the heart. The blood in each of the heart’s four chambers comes briefly to rest before it is ejected by contraction of the heart muscle

If the aorta (diameter d_a) branches into two equal-sized arteries with a combined area equal to that of the aorta, what is the diameter of one of the branches? (a)$\sqrt{\mathbf{d}}$; (b)$\frac{\mathbf{d}}{\sqrt{\mathbf{2}}}$; (c)$\mathbf{2}\mathbf{d}$; (d) $\frac{{\mathbf{d}}_{\mathbf{a}}}{\mathbf{2}}$

The velocity of blood in the aorta can be measured directly with ultrasound techniques. A typical graph of blood velocity versus time during a single heartbeat is shown in Fig. P2.92. Which statement is the best interpretation of this graph? (a) The blood flow changes direction at about 0.25 s; (b) the speed of the blood flow begins to decrease at about 0.10 s; (c) the acceleration of the blood is greatest in magnitude at about 0.25 s; (d) the acceleration of the blood is greatest in magnitude at about 0.10 s.

Question: In your physics lab you release a small glider from rest at various points on a long, frictionless air track that is inclined at an angle above the horizontal. With an electronic photocell, you measure the time it takes the glider to slide a distance from the release point to the bottom of the track. Your measurements are given in Fig. P2.84, which shows a second-order polynomial (quadratic) fit to the plotted data. You are asked to find the glider’s acceleration, which is assumed to be constant. There is some error in each measurement, so instead of using a single set of x and t values, you can be more accurate if you use graphical methods and obtain your measured value of the acceleration from the graph. (a) How can you re-graph the data so that the data points fall close to a straight line? (Hint: You might want to plot or , or both, raised to some power.) (b) Construct the graph you described in part (a) and find the equation for the straight line that is the best fit to the data points. (c) Use the straight line fit from part (b) to calculate the acceleration of the glider. (d) The glider is released at a distance x = 1.35 from the bottom of the track. Use the acceleration value you obtained in part (c) to calculate the speed of the glider when it reaches the bottom of the track.

A simple pendulum (a mass swinging at the end of a string) swings back and forth in a circular arc. What is the direction of the acceleration of the mass when it is at the ends of the swing? At the midpoint? In each case, explain how you obtained your answer 

Redraw Fig. 3.11a if $\overrightarrow{\mathbf{a}}$ is antiparallel to $\overrightarrow{\mathbf{v}}$. Does the particle move in a straight line? What happens to its speed? 

A projectile moves in a parabolic path without air resistance. Is there any point at which $\overrightarrow{\mathbf{a}}$ is parallel to $\overrightarrow{\mathbf{v}}$? Perpendicular to $\overrightarrow{\mathbf{v}}$? Explain

A book slides off a horizontal tabletop. As it leaves the table’s edge, the book has a horizontal velocity of magnitude ${\mathit{v}}_{\mathbf{0}}$. The book strikes the floor in time t. If the initial velocity of the book is doubled to $\mathbf{2}{\mathit{v}}_{\mathbf{0}}$, what happens to (a) the time the book is in the air, (b) the horizontal distance the book travels while it is in the air, and (c) the speed of the book just before it reaches the floor? In particular, does each of these quantities stay the same, double, or change in another way? Explain.

At the instant that you fire a bullet horizontally from a rifle, you drop a bullet from the height of the gun barrel. If there is no air resistance, which bullet hits the level ground first? Explain. 

A package falls out of an airplane that is flying in a straight line at a constant altitude and speed. If you ignore air resistance,what would be the path of the package as observed by the pilot? As observed by a person on the ground?

Sketch the six graphs of the x- and y-components of position, velocity, and acceleration versus time for projectile motion with ${\mathit{x}}_{\mathbf{0}}\mathbf{=}{\mathit{y}}_{\mathbf{0}}\mathbf{=}\mathbf{0}\mathbf{ }$and $\mathbf{0}\mathbf{<}{\mathit{\alpha }}_{\mathbf{0}}\mathbf{<}\mathbf{90}\mathbf{°}$.

If a jumping frog can give itself the same initial speed regardless of the direction in which it jumps (forward or straight up), how is the maximum vertical height to which it can jump related to its maximum horizontal range ${\mathit{R}}_{\mathbf{m}\mathbf{a}\mathbf{x}}\mathbf{=}\frac{{\mathbf{v}}_{\mathbf{0}}^{\mathbf{2}}}{\mathbf{g}}$? 

projectile is fired upward at an angle $\mathit{\theta }$ above the horizontal with an initial speed ${\mathit{v}}_{\mathbf{0}}$ . At its maximum height, what are its velocity vector, its speed, and its acceleration vector? 

In uniform circular motion, what are the average velocity and average acceleration for one revolution? Explain. 

In uniform circular motion, how does the acceleration change when the speed is increased by a factor of 3? When the radius is decreased by a factor of 2?

In uniform circular motion, the acceleration is perpendicular to the velocity at every instant. Is this true when the motion is not uniformthat is, when the speed is not constant? 

Raindrops hitting the side windows of a car in motion often leave diagonal streaks even if there is no wind. Why? Is the explanation the same or different for diagonal streaks on the windshield?

A squirrel has x- and y-coordinates $\left(1.1m, 3.4 m\right)$, at time ${\mathit{t}}_{\mathbf{1}}\mathbf{=}\mathbf{0}$ and coordinates $\mathbf{(}\mathbf{5}\mathbf{.}\mathbf{3}\mathbf{m}\mathbf{,}\mathbf{ }\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{5}\mathbf{ }\mathbf{m}\mathbf{)}$ at time $\mathbf{ }{\mathbf{t}}_{\mathbf{2}}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{s}$. For this time interval, find 

(a) the components of the average velocity, and 

(b) the magnitude and direction of the average velocity.

A rhinoceros is at the origin of coordinates at time ${\mathit{t}}_{\mathbf{1}}\mathbf{=}\mathbf{0}$. For the time interval from ${\mathit{t}}_{\mathbf{1}}\mathbf{=}\mathbf{0}$ to ${\mathit{t}}_{\mathbf{2}}\mathbf{=}\mathbf{12}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{s}$, the rhino’s average velocity has x-component - 3.8 m/s and y-component 4.9 m/s . At time ${\mathit{t}}_{\mathbf{2}}\mathbf{=}\mathbf{12}\mathbf{ }\mathit{s}$, 

(a) what are the x and y-coordinates of the rhino? 

(b) How far is the rhino from the origin?

A web page designer creates an animation in which a dot on a computer screen has position $\overrightarrow{\mathbf{r}}\mathbf{=}\mathbf{[}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{c}\mathbf{m}\mathbf{+}\mathbf{(}\mathbf{2}\mathbf{.}\mathbf{5}\mathbf{ }\mathbf{c}\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}\mathbf{)}{\mathbf{t}}^{\mathbf{2}}\mathbf{]}\hat{\mathbf{i}}\mathbf{+}\mathbf{(}\mathbf{5}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{c}\mathbf{m}\mathbf{/}\mathbf{s}\mathbf{)}\mathit{t}\hat{\mathbf{j}}$. 

(a) Find the magnitude and direction of the dot’s average velocity between $\mathit{t}\mathbf{=}\mathbf{0}$ and $\mathit{t}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{s}$.

(b) Find the magnitude and direction of the instantaneous velocity at $\mathit{t}\mathbf{=}\mathbf{0}\mathbf{ }\mathit{s}$, $\mathit{t}\mathbf{=}\mathbf{0}\mathbf{ }\mathit{s}$, and $\mathit{t}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{s}$. 

(c) Sketch the dot’s trajectory from $\mathit{t}\mathbf{=}\mathbf{0}\mathbf{ }\mathit{s}$ to $\mathit{t}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{s}$, and show the velocities calculated in part (b).

The position of a squirrel running in a park is given by

$\overrightarrow{\mathbf{r}}\mathbf{=}\left[\left(0.0280 m/s\right)t+\left(0.0360 m/s^2\right)t^2\right]\hat{\mathbf{i}}\mathbf{+}\left(0.0190 m/s^3\right){\mathit{t}}^{\mathbf{3}}\hat{\mathbf{j}}$ 

(a) What are ${\mathit{v}}_{\mathbf{x}}\mathit{t}$ and ${\mathit{v}}_{\mathbf{y}}\mathit{t}$, the x- and y-components of the velocity of the squirrel, as functions of time? 

(b) At $\mathit{t}\mathbf{=}\mathbf{5}\mathbf{ }\mathit{s}$, how far is the squirrel from its initial position? 

(c) At $\mathit{t}\mathbf{=}\mathbf{5}\mathbf{ }\mathit{s}$, what are the magnitude and direction of the squirrel’s velocity?

A jet plane is flying at a constant altitude. At time ${\mathit{t}}_{\mathbf{1}}\mathbf{=}\mathbf{0}\mathbf{ }\mathit{s}$, it has components of velocity ${\mathit{v}}_{\mathbf{x}}\mathbf{=}\mathbf{90}\mathbf{ }\mathit{m}\mathbf{/}\mathit{s}$, ${\mathit{v}}_{\mathbf{y}}\mathbf{=}\mathbf{110}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{s}$. At time ${\mathit{t}}_{\mathbf{2}}\mathbf{=}\mathbf{30}\mathbf{ }\mathit{s}$, the components are ${\mathit{v}}_{\mathbf{x}}\mathbf{=}\mathbf{-}\mathbf{170}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{s}$, ${\mathbf{v}}_{\mathbf{y}}\mathbf{=}\mathbf{40}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{s}$. 

(a) Sketch the velocity vectors at ${\mathit{t}}_{\mathbf{1}}$ and ${\mathit{t}}_{\mathbf{2}}$. How do these two vectors differ? For this time interval calculate 

(b) the components of the average acceleration, and 

(c) the magnitude and direction of the average acceleration

A dog running in an open field has components of velocity ${\mathbf{v}}_{\mathbf{x}}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{6}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{s}$ and ${\mathbf{v}}_{\mathbf{y}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{8}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{s}$ at ${\mathit{t}}_{\mathbf{1}}\mathbf{=}\mathbf{10}\mathbf{ }\mathit{s}$. For the time interval from ${\mathit{t}}_{\mathbf{1}}\mathbf{=}\mathbf{10}\mathbf{ }\mathit{s}$ to ${\mathit{t}}_{\mathbf{2}}\mathbf{=}\mathbf{20}\mathbf{ }\mathit{s}$, the average acceleration of the dog has magnitude $\mathbf{0}\mathbf{.}\mathbf{45}\mathbf{ }\mathit{m}\mathbf{/}{\mathit{s}}^{\mathbf{2}}$ and direction 31.0° measured from the +x-axis toward the +y-axis. At ${\mathit{t}}_{\mathbf{2}}\mathbf{=}\mathbf{20}\mathbf{ }\mathit{s}$, (a) what are the x- and y-components of the dog’s velocity? (b) What are the magnitude and direction of the dog’s velocity? (c) Sketch the velocity vectors at ${\mathit{t}}_{\mathbf{1}}$ and ${\mathit{t}}_{\mathbf{2}}$. How do these two vectors differ?

C The coordinates of a bird flying in the -plane are given by $\mathit{x}\left(t\right)\mathbf{=}\mathit{\alpha }\mathit{t}$ and $\mathit{y}\left(t\right)\mathbf{=}\mathbf{3}\mathbf{ }\mathit{m}\mathbf{-}\mathit{\beta }{\mathit{t}}^{\mathbf{2}}$, where $\mathit{\alpha }\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{4}\mathbf{ }\mathit{m}\mathbf{/}\mathit{s}$ and $\mathit{\beta }\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{2}\mathbf{ }\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$ .(a) Sketch the path of the bird between $\mathit{t}\mathbf{=}\mathbf{0}\mathbf{ }\mathit{s}$ and $\mathit{t}\mathbf{=}\mathbf{2}\mathbf{ }\mathit{s}$. (b) Calculate the velocity and acceleration vectors of the bird as functions of time. (c) Calculate the magnitude and direction of the bird’s velocity and acceleration at $\mathit{t}\mathbf{=}\mathbf{2}\mathbf{ }\mathit{s}$. (d) Sketch the velocity and acceleration vectors at $\mathit{t}\mathbf{=}\mathbf{2}\mathbf{ }\mathit{s}$ . At this instant, is the bird’s speed increasing, decreasing, or not changing? Is the bird turning? If so, in what direction?

A remote-controlled car is moving in a vacant parking lot. The velocity of the car as a function of time is given by 

$\overrightarrow{\mathbf{v}}\mathbf{=}\left[5.0 m/s-\left(0.0180 m/s^3\right)t^2\right]\hat{\mathbf{i}}\mathbf{+}\left(2.0 m/s+0.550 m/s^{2}t\right)\hat{\mathbf{j}}$.

(a) What are ${\mathit{a}}_{\mathbf{x}}\left(t\right)$ and ${\mathit{a}}_{\mathbf{y}}\left(t\right)$ , the x- and y-components of the car’s velocity as functions of time? 

(b) What are the magnitude and direction of the car’s velocity at $\mathit{t}\mathbf{=}\mathbf{8}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{s}$? 

(c) What are the magnitude and direction of the car’s acceleration at $\mathit{t}\mathbf{=}\mathbf{8}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{s}$ ?

A physics book slides off a horizontal tabletop with a speed of 1.10 m/s. It strikes the floor in 0.480 s. Ignore air resistance. Find (a) the height of the tabletop above the floor; (b) the horizontal distance from the edge of the table to the point where the book strikes the floor; (c) the horizontal and vertical components of the book’s velocity, and the magnitude and direction of its velocity, just before the book reaches the floor. (d) Draw x-t, y-t, ${\mathit{v}}_{\mathbf{x}}\mathbf{-}\mathit{t}$, and ${\mathit{v}}_{\mathbf{y}}\mathbf{-}\mathit{t}$ graphs for the motion.