Find the following for path A in Figure 3.54: (a) the total distance traveled, and (b) the magnitude and direction of the displacement from start to finish.

In Figure 3.54 the various lines represent paths taken by different people walking in a city. All blocks are 120 m on a side.

Find the north and east components of the displacement for the hikers shown in Figure 3.52.

Suppose you first walk 12.0 m in a direction west of north and then 20.0 m in a direction $40.0^\circ$ south of west. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements A and B , as in Figure 3.56, then this problem finds their sum R = A + B.)

(a) Repeat the problem two problems prior, but for the second leg you walk 20.0 m in a direction $40.0^\circ$ north of east (which is equivalent to subtracting B from A — that is, to finding R' = A - B).

(b) Repeat the problem two problems prior, but now you first walk 20.0 m in a direction $40.0^\circ$ south of west and then 12.0 m in a direction $20.0^\circ$ east of south (which is equivalent to subtracting A from B — that is, to finding R'' = B - A = - R'). Show that this is the case.

Show that the sum of the vectors discussed in Example 3.2 gives the result shown in Figure 3.24.

Example 3.2: A woman sailing a boat at night is following directions to a dock. The instructions read to first sail 27.5 m in a direction $66.0^\circ$ north of east from her current location, and then travel 30.0 m in a direction $112^\circ$ north of east (or $22.0^\circ$ west of north). Find the location of the dock.

Find the following for path C in Figure 3.58: (a) the total distance traveled and (b) the magnitude and direction of the displacement from start to finish. In this part of the problem, explicitly show how you follow the steps of the analytical method of vector addition.

Find the north and east components of the displacement from San Francisco to Sacramento shown in Figure 3.59.

Exercise 3.16: Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements A and B, as in Figure 3.60, then this problem asks you to find their sum R = A + B.)

In an attempt to escape his island, Gilligan builds a raft and sets to sea. The wind shifts a great deal during the day, and he is blown along the following straight lines: 2.50 km $45.0^\circ$ north of west; then 4.70 km $60.0^\circ$ south of east; then 1.30 km $25.0^\circ$ south of west; then 5.10 km straight east; then 1.70 km $5.00^\circ$ east of north; then 7.20 km $55.0^\circ$ south of west; and finally 2.80 km $10.0^\circ$ north of east. What is his final position relative to the island?

A projectile is launched at ground level with an initial speed of 50.0 m/s at an angle of $30.0^\circ$ above the horizontal. It strikes a target above the ground 3.00 seconds later. What are the x and y  distances from where the projectile was launched to where it lands?

A ball is thrown horizontally from the top of a 60.0-m building and lands 100.0 m from the base of the building. Ignore air resistance. (a) How long is the ball in the air? (b) What must have been the initial horizontal component of the velocity? (c) What is the vertical component of the velocity just before the ball hits the ground? (d) What is the velocity (including both the horizontal and vertical components) of the ball just before it hits the ground?

An archer shoots an arrow at a 75.0 m distant target; the bull's-eye of the target is at same height as the release height of the arrow. (a) At what angle must the arrow be released to hit the bull's-eye if its initial speed is 35.0 m/s? In this part of the problem, explicitly show how you follow the steps involved in solving projectile motion problems. (b) There is a large tree halfway between the archer and the target with an overhanging horizontal branch 3.50 m above the release height of the arrow. Will the arrow go over or under the branch?

Verify the ranges for the projectiles in Figure 3.41(a) for and the given initial velocities.

The cannon on a battleship can fire a shell a maximum distance of 32.0 km. (a) Calculate the initial velocity of the shell. (b) What maximum height does it reach? (At its highest, the shell is above 60% of the atmosphere—but air resistance is not really negligible as assumed to make this problem easier.) (c) The ocean is not flat, because the Earth is curved. Assume that the radius of the Earth is $6.37 \times 10^3 \textrm{ km}$. How many meters lower will its surface be 32.0 km from the ship along a horizontal line parallel to the surface at the ship? Does your answer imply that error introduced by the assumption of a flat Earth in projectile motion is significant here?

In the standing broad jump, one squats and then pushes off with the legs to see how far one can jump. Suppose the extension of the legs from the crouch position is 0.600 m and the acceleration achieved from this position is 1.25 times the acceleration due to gravity, $g$ . How far can they jump? State your assumptions. (Increased range can be achieved by swinging the arms in the direction of the jump.)

Serving at a speed of 170 km/h, a tennis player hits the ball at a height of 2.5 m and an angle $\theta$ below the horizontal. The baseline from which the ball is served is 11.9 m from the net, which is 0.91 m high. What is the angle $\theta$ such that the ball just crosses the net? Will the ball land in the service box, which has an outermost service line that is 6.40 m from the net?

Gun sights are adjusted to aim high to compensate for the effect of gravity, effectively making the gun accurate only for a specific range. (a) If a gun is sighted to hit targets that are at the same height as the gun and 100.0 m away, how low will the bullet hit if aimed directly at a target 150.0 m away? The muzzle velocity of the bullet is 275 m/s. (b) Discuss qualitatively how a larger muzzle velocity would affect this problem and what would be the effect of air resistance.

An owl is carrying a mouse to the chicks in its nest. Its position at that time is 4.00 m west and 12.0 m above the center of the 30.0 cm diameter nest. The owl is flying east at 3.50 m/s at an angle $30.0^\circ$ below the horizontal when it accidentally drops the mouse. Is the owl lucky enough to have the mouse hit the nest? To answer this question, calculate the horizontal position of the mouse when it has fallen 12.0 m.

Can a goalkeeper at her/ his goal kick a soccer ball into the opponent's goal without the ball touching the ground? The distance will be about 95 m. A goalkeeper can give the ball a speed of 30 m/s.

A football player punts the ball at a $45.0^\circ$ angle. Without an effect from the wind, the ball would travel 60.0 m horizontally. (a) What is the initial speed of the ball? (b) When the ball is near its maximum height it experiences a brief gust of wind that reduces its horizontal velocity by 1.50 m/s. What distance does the ball travel horizontally?

A seagull flies at a velocity of 9.00 m/s straight into the wind. (a) If it takes the bird 20.0 min to travel 6.00 km relative to the Earth, what is the velocity of the wind? (b) If the bird turns around and flies with the wind, how long will he take to return 6.00 km? (c) Discuss how the wind affects the total round-trip time compared to what it would be with no wind.

Verify that the coin dropped by the airline passenger in the Example 3.8 travels 144 m horizontally while falling 1.50 m in the frame of reference of the Earth.

A ship sets sail from Rotterdam, The Netherlands, heading due north at 7.00 m/s relative to the water. The local ocean current is 1.50 m/s in a direction $40.0^\circ$  north of east. What is the velocity of the ship relative to the Earth?

(a) In what direction would the ship in Exercise 3.57 have to travel in order to have a velocity straight north relative to the Earth, assuming its speed relative to the water remains 7.00 m/s? (b) What would its speed be relative to the Earth?

A sandal is dropped from the top of a 15.0-m-high mast on a ship moving at 1.75 m/s due south. Calculate the velocity of the sandal when it hits the deck of the ship: (a) relative to the ship and (b) relative to a stationary observer on shore. (c) Discuss how the answers give a consistent result for the position at which the sandal hits the deck.

The great astronomer Edwin Hubble discovered that all distant galaxies are receding from our Milky Way Galaxy with velocities proportional to their distances. It appears to an observer on the Earth that we are at the center of an expanding universe. Figure 3.64 illustrates this for five galaxies lying along a straight line, with the Milky Way Galaxy at the center. Using the data from the figure, calculate the velocities: (a) relative to galaxy 2 and (b) relative to galaxy 5. The results mean that observers on all galaxies will see themselves at the center of the expanding universe, and they would likely be aware of relative velocities, concluding that it is not possible to locate the center of expansion with the given information.

An athlete crosses a 25-m-wide river by swimming perpendicular to the water current at a speed of 0.5 m/s relative to the water. He reaches the opposite side at a distance 40 m downstream from his starting point. How fast is the water in the river flowing with respect to the ground? What is the speed of the swimmer with respect to a friend at rest on the ground?

An ice hockey player is moving at 8.00 m/s when he hits the puck toward the goal. The speed of the puck relative to the player is 29.0 m/s. The line between the center of the goal and the player makes a $90.0^\circ$ angle relative to his path as shown in Figure 3.65. What angle must the puck's velocity make relative to the player (in his frame of reference) to hit the center of the goal?

A commercial airplane has an air speed of 280 m/s due east and flies with a strong tailwind. It travels 3000 km in a direction $5^\circ$ south of east in 1.50 h. (a) What was the velocity of the plane relative to the ground? (b) Calculate the magnitude and direction of the tailwind's velocity. (c) What is unreasonable about both of these velocities? (d) Which premise is unreasonable?

A ball is thrown at an angle of 45 degrees above the horizontal. Which of the following best describes the acceleration of the ball from the instant after it leaves the thrower's hand until the time it hits the ground?

1. Always in the same direction as the motion, initially positive and gradually dropping to zero by the time it hits the ground

2. Initially positive in the upward direction, then zero at maximum height, then negative from there until it hits the ground

3. Always in the opposite direction as the motion, initially positive and gradually dropping to zero by the time it hits the ground

4. Always in the downward direction with the same constant value

1. A negative value that does not change with time
2. A gradually increasing negative value (straight line)
3. An increasing rate of negative values over time
4. (parabolic curve)
5. Zero at all times since the initial motion is horizontal

A student wishes to design an experiment to show that the acceleration of an object is independent of the object's velocity. To do this, ball A is launched horizontally with some initial speed at an elevation 1.5 meters above the ground, ball B is dropped from rest 1.5 meters above the ground, and ball C is launched vertically with some initial speed at an elevation 1.5 meters above the ground. What information would the student need to collect about each ball in order to test the hypothesis?

1. Always positive, steadily decreasing
2. Always positive, constant
3. Initially positive, steadily decreasing, becoming negative at the end
4. Initially zero, steadily getting more and more negative

A ball is launched at an angle of 60 degrees above the horizontal, and the vertical position of the ball is recorded at various points in time in the table shown, assuming the ball was at a height of 0 at time t = 0.

a. Draw a graph of the ball's vertical velocity versus time. b. Describe the graph of the ball's horizontal velocity.
c. Draw a graph of the ball's vertical acceleration versus time.