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Relative Motion

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Relative velocity is the velocity of the object after deducting the velocity of the observer.

To determine relative velocity, first choose a frame of reference (a fixed point and a set of directions) and measure velocities relative to the fixed point. Usually the reference point is the ground and the directions are compass points.

EXAMPLE 1

An oarsman can row his boat 3 mph in still water. He sets out on the Illinois River, which flows at 5 mph. We are interested in what an observer on shore measures. When the man heads the boat directly downstream and rows as fast as he can, which direction does the observer on shore see the boat going? When the man heads the boat directly downstream and rows as fast as he can, how fast does the observer on shore see the boat going?

ANSWER TO EXAMPLE 1

velocity relative to shore = velocity of water relative to shore + velocity of boat relative to water

velocity relative to shore = 5 mph [downstream] + 3 mph [downstream] = 8 mph [downstream]

EXAMPLE 2

An aircraft has a speed and direction relative to the air (wind) in which it is traveling, and the wind has a speed and direction relative to the ground. How do you determine the velocity of the aircraft relative to the ground?

ANSWER TO EXAMPLE 2

Ground Velocity = Airspeed and heading + Wind Speed and heading

EXAMPLE 3

An observer sitting on shore sees a canoe traveling 5.0 m/s east, and a sailboat traveling 15.0 m/s west. What is the velocity of the sailboat as observed on the canoe?

ANSWER TO EXAMPLE 3

R = vobject - vobserver

R = vsailboat - vcanoe

R = 15.0 [W] - 5.0 [E]

R = 15.0 [W] + 5.0 {W}

R = 20.0 m/s west

Quick Question

When an airplane starts to fly into a wind, its speed relative to the ground decreases, and when it starts to fly with a wind its ground speed increases. What happens to the ground speed of an airplane if a wind starts to blow at 90 degrees to the direction it is pointing? Its ground speed
a. decreases
b. stays the same
c. increases
d. could do any of the above

ANSWER TO QUICK QUESTION

c. increases Remember to apply vectors to solve the problem.

Relative Motion Problems

(See bottom of page for answers.)

1. A barge that can travel 2.5 m/s on still water is on a river flowing east at 2.0 m/s. What is the velocity of this barge relative to the shore when the barge is heading 30. 0 degrees West of North?

2. A snorkeler who can swim 1.00 m/s in still water aims her body directly across a 150-m-wide river. The current is 0.80 m/s.

a. How far downstream ( from a point opposite her starting point) will she land?

b. How long will it take her to reach the other side?

3. A catamaran whose speed in still water is 5.0 m/s heads west across an estuary. The current is 2.5 m/s south. a. What is the velocity of the catamaran relative to the shore? b. If the estuary is 2395 m wide, how long does it take the catamaran to cross the estuary?

4. A wherry crosses a stream with a velocity relative to the water of 3.30 mi/h at an angle 62.5 degrees north of west. The stream is 0.505 mi wide and has an eastward current of 1.25 mi/h. How far upstream is the wherry when it reaches the opposite shore?

5. Mario skates south at 6.3 m/s. A fellow hockey player passes the puck to him. The puck moves at 10.4 m/s 30° west of south. What are the magnitude and direction of the puck's velocity, as observed by Mario?

6. A runabout can travel 30.0 km/h in still water. How long will it take to arrive 12 km downstream in a river flowing 6.0 km/h?

7. What are the differences between rest and motion?

8. Two dugouts maintain the same speed relative to the water. One dugout travels directly upstream, whereas the other travels directly downstream. An observer determines the velocities of the two dugouts to be -1.2 m/s and +2.9 m/s relative to the shore. What is the speed of the water relative to the shore and what is the speed of each dugout relative to the water?

9. Two rays approach head-on. Ray 1 is traveling with a velocity of 50 zarods per second, ray 2 maintains 60 zarods per second. They start exactly 3,000 zarods apart. How far apart will the two rays be one second before they meet?

10. An observer sitting on shore sees a canoe traveling 5.0 m/s 20° north of east, and a sailboat traveling 15.0 m/s 30° west of north. What is the velocity of the sailboat as observed on the canoe (relative to the canoe)?

11. The passenger in a car observes a telephone pole appearing to move 20.0 m/s to the west, while a cyclist appears to be moving 22.4 m/s 37° north of west. What is the velocity of the cyclist relative to the pole?

12. To the driver in car A, car B appears to be traveling 10 km/h east. To the driver in car B, car C appears to be traveling 110 km/h west. What velocity will car A appear to have according to the driver in car C?

Answers to Relative Motion Problems

Selected solutions are printed below.
For solutions to all the problems on this page click here.

Relative velocity is the vector sum of velocities.

1. Velocity relative to ground = {velocity of barge relative to water} + {velocity of water relative to ground}

Resultant relative velocity is shown.

Add the two vectors using cosine law, or using components as shown below:

Find the sum of the components of the vectors in the E-W axis and in the N-S axis.

Finding relative velocity can be achieved by adding components.
The resultants in each direction are related by Pythagorean Theorem.

Add the resultant components together:

The velocity of this barge with reference to a point on the shore is 2.30 m/s 19.1º East of North.

For solutions to all the problems on this page click here.

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