Relative Motion

Relative Motion with Vectors

All motion is relative

There is no such thing as an object completely at rest.

From some perspective any object can be seen as moving. You may think that your house is at rest, but from the point of view of someone in the space shuttle, your house is rotating with the Earth. And the earth is orbiting the sun. And the sun is orbiting the center of the galaxy. And the galaxy is moving. . . .

Thus to completely describe an object's velocity, you must state the perspective - the point of view - from which the velocity is measured. The phrase point of view is sometimes called a frame of reference.

The most common frame of reference for describing motion is from the surface of the Earth. The moving platform and bouncing ball shown below are seen from the frame of reference of someone at rest on the surface of the Earth.

This same motion can also be described from the frame of reference of the moving platform. In this case, the earth is moving and the platform is seen to be at rest.

Frame of reference rules

The frame of reference from which motion is measured is always taken to be at rest. In other words, the observer always sees himself to be at rest and the world moves around him.
Two objects moving in different frames of reference will see each other moving with opposite velocities: v_a/b = −v_b/a
To shift from one frame of reference to another, use a Galilean transformation, described in the next section.

Galilean Transformations

A Galilean transformation is a simple algebraic shift to move a measurement from one frame of reference to another. The formula to do this is:

In words, this formula can be stated as: the velocity of object A with respect to C is equal to the velocity of A with respect to B plus the velocity of B with respect to C.

Play with the applet that follows then examine the example below it to see how to make a Galilean transformation.

Example 1 - a Galilean highway

A) Find the speed of the car, truck and earth with respect to you.
B) Find the speed of you, the truck and the earth with respect to the car.
C) Find the speed of you, the car and the earth with respect to the truck.

Relative motion in 2-dimensions

The same equation for making a Galilean transformation applies when considereing 2-dimensional motion. However, you must consider the equation to be a vector addition equation now. Apply the rules of vector addition you learned in Unit 1.

Example 2 - a westerly wind

On a windless day, Moustapha Jones's helicopter can fly at a top speed of 275km/hr. If he heads straight north at top speed, but then a wind blows west at 80 km/hr, what is Moustapha's velocity with respect to the earth?

Example 3 - streaming media

Moustapha Jones has a party boat that can travel in still water at a top speed of 65 km/hr. He needs to travel from the east bank directly across a river that is flowing from north to south at a speed of 24 km/hr.
A) In what direction should he point his boat with the motor at full throttle, to go straight across the river?
B) What will be his final velocity as he crosses the river?