Relative Motion

How do I determine relative velocity?

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

Questions:

1. A boat that can travel 2.5 m/s on still water is now on a river that flows due east at a velocity of 2.0 m/s. What is the velocity of this boat with reference to a point on the shore when the boat is headed 30. 0 degrees West of North.

2. A swimmer is capable of swimming 1.00 m/s in still water. If she aims her body directly across a 150-m-wide river whose 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 boat whose speed in still water is 5.0 m/s is headed west across a river. The river current is 2.5 m/s south. a. What is the velocity of the boat relative to the shore? b. If the river is 2395 m wide, how long does it take the boat to cross the river?

4. In radio controlled model airplane competition, each plane must fly from the center of a 1km radius circle to any point on the circle and back to the center. The winner is the plane with the shortest round-trip time. The contestants are free to fly their planes along any route so long as the plane begins at the center, travels to the circle and then returns to the center. On the day of the race, a steady wind blows out of the north at 5m/s. Your plane can maintain an airspeed of 15m/s. Strategy is paramount. Should you fly your plane upwind on the first leg and downwind on the trip back or across the wind flying east and then west? Optimize your chances by calculating the round-trip time for both routes.

5. Mario, a hockey player, is skating due south at a speed of 6.3 m/s relative to the ice. A teammate passes the puck to him. The puck has a speed of 10.4 m/s and is moving in a direction of 30° west of south, relative to the ice. What are the magnitude and direction (relative to due south) of the puck's velocity, as observed by Mario?

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

7. State the differences between rest and motion.

8. Two canoeists in identical canoes exert maintain the same speed relative to the water. One paddles directly upstream (and moves upstream), whereas the other paddles directly downstream. With downstream as the positive direction, an observer on shore determines the velocities of the two canoes to be -1.2 m/s and +2.9 m/s, respectively. What is the speed of the water relative to the shore and what is the speed of each canoe relative to the water?

Answers:

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

Relative velocity is the vector sum of velocities.

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

Resultant relative velocity is shown.

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

Finding relative velocity can be achieved by adding components.

Add the resultant components together:

The resultants in each direction are related by Pythagorean Theorem.

The velocity of this boat 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.