# 3.5 Addition of velocities  (Page 6/12)

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Near the end of a marathon race, the first two runners are separated by a distance of 45.0 m. The front runner has a velocity of 3.50 m/s, and the second a velocity of 4.20 m/s. (a) What is the velocity of the second runner relative to the first? (b) If the front runner is 250 m from the finish line, who will win the race, assuming they run at constant velocity? (c) What distance ahead will the winner be when she crosses the finish line?

(a) 0.70 m/s faster

(b) Second runner wins

(c) 4.17 m

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

A football quarterback is moving straight backward at a speed of 2.00 m/s when he throws a pass to a player 18.0 m straight downfield. The ball is thrown at an angle of $25.0º$ relative to the ground and is caught at the same height as it is released. What is the initial velocity of the ball relative to the quarterback ?

$17\text{.}\text{0 m/s}$ , $\text{22}\text{.}1º$

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 $\text{40.0º}$ north of east. What is the velocity of the ship relative to the Earth?

(a) A jet airplane flying from Darwin, Australia, has an air speed of 260 m/s in a direction $5.0º$ south of west. It is in the jet stream, which is blowing at 35.0 m/s in a direction $15º$ south of east. What is the velocity of the airplane relative to the Earth? (b) Discuss whether your answers are consistent with your expectations for the effect of the wind on the plane’s path.

(a) $2\text{30 m/s}$ , $8.0º$ south of west

(b) The wind should make the plane travel slower and more to the south, which is what was calculated.

(a) In what direction would the ship in [link] have to travel in order to have a velocity straight north relative to the Earth, assuming its speed relative to the water remains $7\text{.}\text{00 m/s}$ ? (b) What would its speed be relative to the Earth?

(a) Another airplane is flying in a jet stream that is blowing at 45.0 m/s in a direction $\text{20º}$ south of east (as in [link] ). Its direction of motion relative to the Earth is $45.0º$ south of west, while its direction of travel relative to the air is $5.00º$ south of west. What is the airplane’s speed relative to the air mass? (b) What is the airplane’s speed relative to the Earth?

(a) 63.5 m/s

(b) 29.6 m/s

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 velocity of the wind relative to the water is crucial to sailboats. Suppose a sailboat is in an ocean current that has a velocity of 2.20 m/s in a direction $30.0º$ east of north relative to the Earth. It encounters a wind that has a velocity of 4.50 m/s in a direction of $50.0º$ south of west relative to the Earth. What is the velocity of the wind relative to the water?

$6\text{.}\text{68 m/s}$ , $\text{53}\text{.}3º$ south of west

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