4.5 Relative motion in one and two dimensions (Page 5/8)

University physics volume 1 Page 5 / 8

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

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A boat can be rowed at 8.0 km/h in still water. (a) How much time is required to row 1.5 km downstream in a river moving 3.0 km/h relative to the shore? (b) How much time is required for the return trip? (c) In what direction must the boat be aimed to row straight across the river? (d) Suppose the river is 0.8 km wide. What is the velocity of the boat with respect to Earth and how much time is required to get to the opposite shore? (e) Suppose, instead, the boat is aimed straight across the river. How much time is required to get across and how far downstream is the boat when it reaches the opposite shore?

Take the positive direction to be the same direction that the river is flowing, which is east. S = shore/Earth, W = water, and B = boat.
a. ${\vec{v}}_{BS} = 11 km / h$
$t = 8.2 min$
b. ${\vec{v}}_{BS} = −5 km / h$
$t = 18 min$
c. ${\vec{v}}_{BS} = {\vec{v}}_{BW} + {\vec{v}}_{WS}$ $θ = 22 °$ west of north
Vectors V sub B W, V sub W S and V sub B S form a right triangle, with V sub B W as the hypotenuse. V sub B S points up. V sub W S points to the right. V sub B W points up and left, at an angle of theta to the vertical. V sub B S is the vector sum of v sub B W and V sub W S.
d. $| {\vec{v}}_{BS} | = 7.4 km / h$ $t = 6.5 min$
e. ${\vec{v}}_{BS} = 8.54 km / h,$ but only the component of the velocity straight across the river is used to get the time
Vectors V sub B W, V sub W S and V sub B S form a right triangle, with V sub B S as the hypotenuse. V sub B W points up. V sub W S points to the right. V sub B S points up and right, at an angle of theta to the vertical. V sub B S is the vector sum of v sub B W and V sub W S.
$t = 6.0 min$
Downstream = 0.3 km

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A small plane flies at 200 km/h in still air. If the wind blows directly out of the west at 50 km/h, (a) in what direction must the pilot head her plane to move directly north across land and (b) how long does it take her to reach a point 300 km directly north of her starting point?

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A cyclist traveling southeast along a road at 15 km/h feels a wind blowing from the southwest at 25 km/h. To a stationary observer, what are the speed and direction of the wind?

${\vec{v}}_{A G} = {\vec{v}}_{A C} + {\vec{v}}_{C G}$
$| {\vec{v}}_{A C} | = 25 km / h | {\vec{v}}_{C G} | = 15 km / h | {\vec{v}}_{A G} | = 29.15 km / h$ ${\vec{v}}_{A G} = {\vec{v}}_{A C} + {\vec{v}}_{C G}$
The angle between ${\vec{v}}_{A C}$ and ${\vec{v}}_{A G}$ is $31 °,$ so the direction of the wind is $14 °$ north of east.
Vectors V sub A C, V sub C G and V sub A G form a triangle. V sub A C and V sub C G are at right angles. V sub A G is the vector sum of v sub A C and V sub C G.

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A river is moving east at 4 m/s. A boat starts from the dock heading $30 °$ north of west at 7 m/s. If the river is 1800 m wide, (a) what is the velocity of the boat with respect to Earth and (b) how long does it take the boat to cross the river?

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Additional problems

A Formula One race car is traveling at 89.0 m/s along a straight track enters a turn on the race track with radius of curvature of 200.0 m. What centripetal acceleration must the car have to stay on the track?

$a_{C} = 39.6 m / s^{2}$

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A particle travels in a circular orbit of radius 10 m. Its speed is changing at a rate of $15.0 m / s^{2}$ at an instant when its speed is 40.0 m/s. What is the magnitude of the acceleration of the particle?

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The driver of a car moving at 90.0 km/h presses down on the brake as the car enters a circular curve of radius 150.0 m. If the speed of the car is decreasing at a rate of 9.0 km/h each second, what is the magnitude of the acceleration of the car at the instant its speed is 60.0 km/h?

$90.0 km / h = 25.0 m / s, 9.0 km / h = 2.5 m / s,$ $60.0 km / h = 16.7 m / s$
$a_{T} = −2.5 {m/s}^{2}, a_{C} = 1.86 m / s^{2}, a = 3.1 m / s^{2}$

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A race car entering the curved part of the track at the Daytona 500 drops its speed from 85.0 m/s to 80.0 m/s in 2.0 s. If the radius of the curved part of the track is 316.0 m, calculate the total acceleration of the race car at the beginning and ending of reduction of speed.

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Source: OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5

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