Chapter 3

A squirrel has \(x\)- and \(y\)-coordinates (1.1 m, 3.4 m) at time \(t_1\) = 0 and coordinates (5.3 m, -0.5 m) at time \(t_2\) = 3.0 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 \(t_1\) = 0. For the time interval from \(t_1\) = 0 to \(t_2\) = 12.0 s, the rhino's average velocity has \(x\)-component -3.8 m/s and y-component 4.9 m/s. At time \(t_2\) = 12.0 s, (a) what are the \(x\)- and \(y\)-coordinates of the rhino? (b) How far is the rhino from the origin?

CALC A web page designer creates an animation in which a dot on a computer screen has position $$ \vec{r} =[34.0 cm +(2.5 cm/s^2)t^2] \hat{i} +(5.0 cm/s)t \hat{\jmath}.$$ (a) Find the magnitude and direction of the dot's average velocity between \(t\) = 0 and \(t\) = 2.0 s.(b) Find the magnitude and direction of the instantaneous velocity at \(t\) = 0, \(t\) = 1.0 s, and \(t\) = 2.0 s. (c) Sketch the dot's trajectory from \(t\) = 0 to \(t\) = 2.0 s, and show the velocities calculated in part (b).

CALC The position of a squirrel running in a park is given by \(\vec{r}= [(0.280 m/s)t + (0.0360 m/s^2)t^2] \hat{\imath}+(0.0190 m/s^3)t^3\hat{\jmath}\). (a) What are \(v_x(t)\) and \(v_y(t)\), the \(x\)- and \(y\)-components of the velocity of the squirrel, as functions of time? (b) At \(t\) = 5.00 s, how far is the squirrel from its initial position? (c) At \(t\) = 5.00 s, what are the magnitude and direction of the squirrel's velocity?

A jet plane is flying at a constant altitude. At time \(t_1\) = 0, it has components of velocity \(v_x\) = 90 m/s, \(v_y\) = 110 m/s. At time \(t_2\) = 30.0 s, the components are \(v_x\) = -170 m/s, \(v_y\) = 40 m/s. (a) Sketch the velocity vectors at \(t_1\) and \(t_2\). 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 \(v_x\) = 2.6 m/s and \(v_y\) = -1.8 m/s at \(t_1\) = 10.0 s. For the time interval from \(t_1\) = 10.0 s to \(t_2\) = 20.0 s, the average acceleration of the dog has magnitude 0.45 m/s\(^2\) and direction 31.0\(^\circ\) measured from the +\(x\)-axis toward the +\(y\)-axis. At \(t_2\) = 20.0 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 \(t_1\) and \(t_2\). How do these two vectors differ?

CALC The coordinates of a bird flying in the \(xy\)-plane are given by \(x(t)\) = \(at\) and \(y(t)\) = 3.0 m - \(\beta t^2\), where \(\alpha\) = 2.4 m/s and \(\beta\) = 1.2 m/s\(^2\). (a) Sketch the path of the bird between \(t\) = 0 and \(t\) = 2.0 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 \(t\) = 2.0 s. (d) Sketch the velocity and acceleration vectors at \(t\) = 2.0 s. At this instant, is the bird's speed increasing, decreasing, or not changing? Is the bird turning? If so, in what direction?

CALC A remote-controlled car is moving in a vacant parking lot. The velocity of the car as a function of time is given by \(\hat{v} =[5.00 m/s - (0.0180 m/s^3)t^2] \hat{\imath} = [2.00 m/s + (0.550 m/s^2)t]\hat{j}\). (a) What are \(a_x(t)\) and \(a_y(t)\), 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 \(t\) = 8.00 s? (b) What are the magnitude and direction of the car's acceleration at \(t\) = 8.00 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 \(x-t, y-t, v_x-t\), and \(v_y-t\) graphs for the motion.

Crickets Chirpy and Milada jump from the top of a vertical cliff. Chirpy drops downward and reaches the ground in 2.70 s, while Milada jumps horizontally with an initial speed of 95.0 cm/s. How far from the base of the cliff will Milada hit the ground? Ignore air resistance.

A rookie quarterback throws a football with an initial upward velocity component of 12.0 m/s and a horizontal velocity component of 20.0 m/s. Ignore air resistance. (a) How much time is required for the football to reach the highest point of the trajectory? (b) How high is this point? (c) How much time (after it is thrown) is required for the football to return to its original level? How does this compare with the time calculated in part (a)? (d) How far has the football traveled horizontally during this time? (e) Draw \(x-t, y-t, v_x-t\), and \(v_y-t\) graphs for the motion.

During a storm, a car traveling on a level horizontal road comes upon a bridge that has washed out. The driver must get to the other side, so he decides to try leaping the river with his car. The side of the road the car is on is 21.3 m above the river, while the opposite side is only 1.8 m above the river. The river itself is a raging torrent 48.0 m wide. (a) How fast should the car be traveling at the time it leaves the road in order just to clear the river and land safely on the opposite side? (b) What is the speed of the car just before it lands on the other side?

The froghopper, \(Philaenus\) \(spumarius\), holds the world record for insect jumps. When leaping at an angle of 58.0\(^\circ\) above the horizontal, some of the tiny critters have reached a maximum height of 58.7 cm above the level ground. (See \(Nature\), Vol. 424, July 31, 2003, p. 509.) (a) What was the takeoff speed for such a leap? (b) What horizontal distance did the froghopper cover for this world-record leap?

Inside a starship at rest on the earth, a ball rolls off the top of a horizontal table and lands a distance \(D\) from the foot of the table. This starship now lands on the unexplored Planet \(X\). The commander, Captain Curious, rolls the same ball off the same table with the same initial speed as on earth and finds that it lands a distance 2.76\(D\) from the foot of the table. What is the acceleration due to gravity on Planet \(X\)?

On level ground a shell is fired with an initial velocity of 40.0 m/s at 60.0\(^\circ\) above the horizontal and feels no appreciable air resistance. (a) Find the horizontal and vertical components of the shell's initial velocity. (b) How long does it take the shell to reach its highest point? (c) Find its maximum height above the ground. (d) How far from its firing point does the shell land? (e) At its highest point, find the horizontal and vertical components of its acceleration and velocity.

A major leaguer hits a baseball so that it leaves the bat at a speed of 30.0 m/s and at an angle of 36.9\(^\circ\) above the horizontal. Ignore air resistance. (a) At what \(two\) times is the baseball at a height of 10.0 m above the point at which it left the bat? (b) Calculate the horizontal and vertical components of the baseball's velocity at each of the two times calculated in part (a). (c) What are the magnitude and direction of the baseball's velocity when it returns to the level at which it left the bat?

Firemen use a high-pressure hose to shoot a stream of water at a burning building. The water has a speed of 25.0 m/s as it leaves the end of the hose and then exhibits projectile motion. The firemen adjust the angle of elevation \(\alpha\) of the hose until the water takes 3.00 s to reach a building 45.0 m away. Ignore air resistance; assume that the end of the hose is at ground level. (a) Find \(\alpha\). (b) Find the speed and acceleration of the water at the highest point in its trajectory. (c) How high above the ground does the water strike the building, and how fast is it moving just before it hits the building?

A man stands on the roof of a 15.0-m-tall building and throws a rock with a speed of 30.0 m/s at an angle of 33.0\(^\circ\) above the horizontal. Ignore air resistance. Calculate (a) the maximum height above the roof that the rock reaches; (b) the speed of the rock just before it strikes the ground; and (c) the horizontal range from the base of the building to the point where the rock strikes the ground. (d) Draw \(x-t, y-t, v_x-t\), and \(v_y-t\) graphs for the motion.

A 124-kg balloon carrying a 22-kg basket is descending with a constant downward velocity of 20.0 m/s. A 1.0-kg stone is thrown from the basket with an initial velocity of 15.0 m/s perpendicular to the path of the descending balloon, as measured relative to a person at rest in the basket. That person sees the stone hit the ground 5.00 s after it was thrown. Assume that the balloon continues its downward descent with the same constant speed of 20.0 m/s. (a) How high is the balloon when the rock is thrown? (b) How high is the balloon when the rock hits the ground? (c) At the instant the rock hits the ground, how far is it from the basket? (d) Just before the rock hits the ground, find its horizontal and vertical velocity components as measured by an observer (i) at rest in the basket and (ii) at rest on the ground.

The earth has a radius of 6380 km and turns around once on its axis in 24 h. (a) What is the radial acceleration of an object at the earth's equator? Give your answer in m/s\(^2\) and as a fraction of \(g\). (b) If \(a_{rad}\) at the equator is greater than \(g\), objects will fly off the earth's surface and into space. (We will see the reason for this in Chapter 5.) What would the period of the earth's rotation have to be for this to occur?

Our balance is maintained, at least in part, by the endolymph fluid in the inner ear. Spinning displaces this fluid, causing dizziness. Suppose that a skater is spinning very fast at 3.0 revolutions per second about a vertical axis through the center of his head. Take the inner ear to be approximately 7.0 cm from the axis of spin. (The distance varies from person to person.) What is the radial acceleration (in m/s\(^2\) and in \(g\)'s) of the endolymph fluid?

A model of a helicopter rotor has four blades, each 3.40 m long from the central shaft to the blade tip. The model is rotated in a wind tunnel at 550 rev/min. (a) What is the linear speed of the blade tip, in m/s? (b) What is the radial acceleration of the blade tip expressed as a multiple of \(g\)?

A Ferris wheel with radius 14.0 m is turning about a horizontal axis through its center (\(\textbf{Fig. E3.27}\)). The linear speed of a passenger on the rim is constant and equal to 6.00 m/s. What are the magnitude and direction of the passenger's acceleration as she passes through (a) the lowest point in her circular motion and (b) the highest point in her circular motion? (c) How much time does it take the Ferris wheel to make one revolution?

The radius of the earth's orbit around the sun (assumed to be circular) is 1.50 \(\times\) 10\(^8\) km, and the earth travels around this orbit in 365 days. (a) What is the magnitude of the orbital velocity of the earth, in m/s? (b) What is the radial acceleration of the earth toward the sun, in m/s\(^2\) ? (c) Repeat parts (a) and (b) for the motion of the planet Mercury (orbit radius = 5.79 \(\times\) 10\(^7\) km, orbital period = 88.0 days).

At its Ames Research Center, NASA uses its large "20-G" centrifuge to test the effects of very large accelerations ("hypergravity") on test pilots and astronauts. In this device, an arm 8.84 m long rotates about one end in a horizontal plane, and an astronaut is strapped in at the other end. Suppose that he is aligned along the centrifuge's arm with his head at the outermost end. The maximum sustained acceleration to which humans are subjected in this device is typically 12.5\(g\). (a) How fast must the astronaut's head be moving to experience this maximum acceleration? (b) What is the \(difference\) between the acceleration of his head and feet if the astronaut is 2.00 m tall? (c) How fast in rpm (rev/min) is the arm turning to produce the maximum sustained acceleration?

A railroad flatcar is traveling to the right at a speed of 13.0 m/s relative to an observer standing on the ground. Someone is riding a motor scooter on the flatcar (\(\textbf{Fig. E3.30}\)). What is the velocity (magnitude and direction) of the scooter relative to the flatcar if the scooter's velocity relative to the observer on the ground is (a) 18.0 m/s to the right? (b) 3.0 m/s to the left? (c) zero?

A "moving sidewalk" in an airport terminal moves at 1.0 m/s and is 35.0 m long. If a woman steps on at one end and walks at 1.5 m/s relative to the moving sidewalk, how much time does it take her to reach the opposite end if she walks (a) in the same direction the sidewalk is moving? (b) In the opposite direction?

Two piers, \(A\) and \(B\), are located on a river; \(B\) is 1500 m downstream from A (\(\textbf{Fig. E3.32}\)). Two friends must make round trips from pier \(A\) to pier \(B\) and return. One rows a boat at a constant speed of 4.00 km/h relative to the water; the other walks on the shore at a constant speed of 4.00 km/h. The velocity of the river is 2.80 km/h in the direction from \(A\) to \(B\). How much time does it take each person to make the round trip?

A canoe has a velocity of 0.40 m/s southeast relative to the earth. The canoe is on a river that is flowing 0.50 m/s east relative to the earth. Find the velocity (magnitude and direction) of the canoe relative to the river.

The nose of an ultralight plane is pointed due south, and its airspeed indicator shows 35 m/s. The plane is in a 10-m/s wind blowing toward the southwest relative to the earth. (a) In a vectoraddition diagram, show the relationship of \(\vec{v}_{P/E}\) (the velocity of the plane relative to the earth) to the two given vectors. (b) Let \(x\) be east and y be north, and find the components of \(\vec{v} _{P/E}\). (c) Find the magnitude and direction of \(\vec{v} _{P/E}\).

A river flows due south with a speed of 2.0 m/s. You steer a motorboat across the river; your velocity relative to the water is 4.2 m/s due east. The river is 500 m wide. (a) What is your velocity (magnitude and direction) relative to the earth? (b) How much time is required to cross the river? (c) How far south of your starting point will you reach the opposite bank?

Canada geese migrate essentially along a north-south direction for well over a thousand kilometers in some cases, traveling at speeds up to about 100 km/h. If one goose is flying at 100 km/h relative to the air but a 40-km/h wind is blowing from west to east, (a) at what angle relative to the north-south direction should this bird head to travel directly southward relative to the ground? (b) How long will it take the goose to cover a ground distance of 500 km from north to south? (\(Note\): Even on cloudy nights, many birds can navigate by using the earth's magnetic field to fix the north-south direction.)

An airplane pilot wishes to fly due west. A wind of 80.0 km/h (about 50 mi/h) is blowing toward the south. (a) If the airspeed of the plane (its speed in still air) is 320.0 km/h (about 200 mi/h), in which direction should the pilot head? (b) What is the speed of the plane over the ground? Draw a vector diagram.

A rocket is fired at an angle from the top of a tower of height \(h_0\) = 50.0 m. Because of the design of the engines, its position coordinates are of the form \(x(t) = A + Bt^2 \)and \(y(t) = C + Dt^3\), where \(A, B, C,\) and \(D\) are constants. The acceleration of the rocket 1.00 s after firing is \(\vec{a} = (4.00 \hat{i}+ 3.00\hat{j}) m/s^2\). Take the origin of coordinates to be at the base of the tower. (a) Find the constants \(A, B, C,\) and \(D\), including their SI units. (b) At the instant after the rocket is fired, what are its acceleration vector and its velocity? (c) What are the \(x\)- and \(y\)-components of the rocket's velocity 10.0 s after it is fired, and how fast is it moving? (d) What is the position vector of the rocket 10.0 s after it is fired?

A faulty model rocket moves in the \(xy\)-plane (the positive \(y\)-direction is vertically upward). The rocket's acceleration has components \(a_x(t) = \alpha t^2\) and \(a_y(t) = \beta - \gamma t\), where \(\alpha = 2.50 m/s^4, \beta = 9.00 m/s^2,\) and \(\gamma = 1.40 m/s^3\). At \(t = 0\) the rocket is at the origin and has velocity \(\vec{v}_0=v_0\hat{i} + v_{0y}\hat{j}\) with \(v_{0x}\) = 1.00 m/s and \(v_{0y}\) = 7.00 m/s. (a) Calculate the velocity and position vectors as functions of time. (b) What is the maximum height reached by the rocket? (c) What is the horizontal displacement of the rocket when it returns to \(y = 0\)?

If \(\vec{r} = bt^2\hat{\imath} + ct^3\hat{\jmath}\), where \(b\) and \(c\) are positive constants, when does the velocity vector make an angle of 45.0\(^\circ\) with the \(x\)- and \(y\)-axes?

The position of a dragonfly that is flying parallel to the ground is given as a function of time by \(\vec{r} = [2.90 m + (0.0900 m/s^2)t^2] \hat{\imath} - (0.0150 m/s^3)t^3\hat{\jmath}\). (a) At what value of \(t\) does the velocity vector of the dragonfly make an angle of 30.0\(^\circ\) clockwise from the \(+x\)-axis? (b) At the time calculated in part (a), what are the magnitude and direction of the dragonfly's acceleration vector?

A bird flies in the \(xy\)-plane with a velocity vector given by \(\overrightarrow{v}\) = \((a - \beta$$t^{2})\) \(\hat{l}+\gamma t \hat{j}\), with \(\alpha = 2.4 m/s\), \(\beta = 1.6 m/s^3\), and \(\gamma = 4.0 m/s^2\). The positive y-direction is vertically upward. At \(t = 0\) the bird is at the origin. (a) Calculate the position and acceleration vectors of the bird as functions of time. (b) What is the bird's altitude (y-coordinate) as it flies over \(x = 0\) for the first time after \(t = 0\)?

A sly 1.5-kg monkey and a jungle veterinarian with a blow-gun loaded with a tranquilizer dart are 25 m above the ground in trees 70 m apart. Just as the veterinarian shoots horizontally at the monkey, the monkey drops from the tree in a vain attempt to escape being hit. What must the minimum muzzle velocity of the dart be for the dart to hit the monkey before the monkey reaches the ground?

Birds of prey typically rise upward on thermals. The paths these birds take may be spiral-like. You can model the spiral motion as uniform circular motion combined with a constant upward velocity. Assume that a bird completes a circle of radius 6.00 m every 5.00 s and rises vertically at a constant rate of 3.00 m/s. Determine (a) the bird's speed relative to the ground; (b) the bird's acceleration (magnitude and direction); and (c) the angle between the bird's velocity vector and the horizontal.

In fighting forest fires, airplanes work in support of ground crews by dropping water on the fires. For practice, a pilot drops a canister of red dye, hoping to hit a target on the ground below. If the plane is flying in a horizontal path 90.0 m above the ground and has a speed of 64.0 m/s (143 mi/h), at what horizontal distance from the target should the pilot release the canister? Ignore air resistance.

A movie stuntwoman drops from a helicopter that is 30.0 m above the ground and moving with a constant velocity whose components are 10.0 m/s upward and 15.0 m/s horizontal and toward the south. Ignore air resistance. (a) Where on the ground (relative to the position of the helicopter when she drops) should the stuntwoman have placed foam mats to break her fall? (b) Draw \(x\)-\(t\), \(y\)-\(t\), \(v$$_x\)-\(t\), and \(v$$_y\)-\(t\) graphs of her motion.

An airplane is flying with a velocity of 90.0 m/s at an angle of 23.0\(^{\circ}\) above the horizontal. When the plane is 114 m directly above a dog that is standing on level ground, a suitcase drops out of the luggage compartment. How far from the dog will the suitcase land? Ignore air resistance.

A cannon, located 60.0 m from the base of a vertical 25.0-m-tall cliff, shoots a 15-kg shell at 43.0\(^{\circ}\) above the horizontal toward the cliff. (a) What must the minimum muzzle velocity be for the shell to clear the top of the cliff? (b) The ground at the top of the cliff is level, with a constant elevation of 25.0 m above the cannon. Under the conditions of part (a), how far does the shell land past the edge of the cliff?

A toy rocket is launched with an initial velocity of 12.0 m/s in the horizontal direction from the roof of a 30.0-m-tall building. The rocket's engine produces a horizontal acceleration of \((1.60 m/s^3)t\), in the same direction as the initial velocity, but in the vertical direction the acceleration is \(g\), downward. Ignore air resistance. What horizontal distance does the rocket travel before reaching the ground?

According to \(Guinness\) \(World\) \(Records\), the longest home run ever measured was hit by Roy "Dizzy" Carlyle in a minor league game. The ball traveled 188 m (618 ft) before landing on the ground outside the ballpark. (a) If the ball's initial velocity was in a direction 45\(^{\circ}\) above the horizontal, what did the initial speed of the ball need to be to produce such a home run if the ball was hit at a point 0.9 m (3.0 ft) above ground level? Ignore air resistance, and assume that the ground was perfectly flat. (b) How far would the ball be above a fence 3.0 m (10 ft) high if the fence was 116 m (380 ft) from home plate?

An airplane is dropping bales of hay to cattle stranded in a blizzard on the Great Plains. The pilot releases the bales at 150 m above the level ground when the plane is flying at 75 m/s in a direction 55\(^{\circ}\) above the horizontal. How far in front of the cattle should the pilot release the hay so that the bales land at the point where the cattle are stranded?

A baseball thrown at an angle of 60.0\(^{\circ}\) above the horizontal strikes a building 18.0 m away at a point 8.00 m above the point from which it is thrown. Ignore air resistance. (a) Find the magnitude of the ball's initial velocity (the velocity with which the ball is thrown). (b) Find the magnitude and direction of the velocity of the ball just before it strikes the building.

A boy 12.0 m above the ground in a tree throws a ball for his dog, who is standing right below the tree and starts running the instant the ball is thrown. If the boy throws the ball horizontally at 8.50 m/s, (a) how fast must the dog run to catch the ball just as it reaches the ground, and (b) how far from the tree will the dog catch the ball?

A rock is thrown with a velocity \(v_0\), at an angle of \(\alpha_0\) from the horizontal, from the roof of a building of height \(h\). Ignore air resistance. Calculate the speed of the rock just before it strikes the ground, and show that this speed is independent of \(\alpha_0\).