Relative Motion

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3.13Relative Motion


In these revision notes for Relative Motion, we cover the following key points:

  • The meaning of relative motion
  • How does the movement of reference frame affects the motion of an object?
  • How do the kinematic quantities change during the relative motion?
  • How to calculate the position of objects in different situations during a relative motion?

Relative Motion Revision Notes

The motion of an object does not depend only on the values of its kinematic quantities but also on the reference frame chosen to study its motion. Therefore, we say "the motion is relative; its parameters may be different in two different reference frames although the rhythm of motion may be the same."

When two objects are moving towards to or away from each other, one of the objects is taken as a reference frame and the other as a normal moving object. Said this, there are six possible cases regarding the objects involved in a relative motion. They are:

Case 1 - Reference frame is at rest and the object is moving at constant velocity

This includes the situations described in Uniform Motion explained in the previous tutorials. Thus, if the initial position of the object is x0 units to the reference point and the object is moving at constant velocity v, the equation of motion which gives the position x of the object at any instant t, is

Equation 1

x(t) = x0 + v × t

Case 2 - Reference frame is moving at constant velocity v' and the object is at rest

In this case, the object's velocity is v = 0 but the reference frame's velocity is v'. If the reference frame is moving let's say due right (in the positive direction), the object seems moving due left (in the negative direction), although it is at rest.

If the object is initially at x0, after t seconds, it will be at

Equation 2

x(t) = x0 + v × t - v' × t

Since the object is at rest (v = 0), the second term of the above equation is cancelled, so we obtain

x(t) = x0 - v' × t

where v' is the velocity by which the reference frame is moving.

Case 3 - Reference frame is moving at constant velocity v' and the object is moving at constant velocity v

This is the case when two objects are moving either in the same or in the opposite direction. One of the objects is taken as a reference frame (for example the car in which we are inside) and the other object is considered as an object moving in respect to the first. Once again, we use the equation (2)

x(t) = x0 + v × t - v' × t

to find the position of the second object in respect to the first one (the reference frame).

Case 4 - Reference point is at rest and the object is moving at constant acceleration a

This case represents the situations described earlier in tutorials involving motion with constant acceleration. Thus, since v' = 0, we obtain for the position of an object at any instant t (if the object which initially was at x0 and it is moving at constant acceleration a):

Equation 3

x(t) = x0 + v0 × t + a × t2/2 - v' × t (3)

or

x(t) = x0 + v0 × t + a × t2/2

since v' = 0 and as a result, the last term of equation (3) is cancelled.

Case 5 - Reference point is moving at constant velocity v and the object is moving at constant acceleration a

In this case, the last term of the equation (3) is not cancelled but it is an active term of the equation. Thus, we have:

x(t) = x0 + v0 × t + a × t2/2 - v' × t

Factorizing, we obtain

Equation 4

x(t) = x0 + (v0 - v') × t + a × t2/2

Case 6 - Both the object and the reference frame are moving at constant acceleration

Depending on the direction of motion of the object and reference frame, we have

Equation 5

x(t) = x0 + v0 × t + a × t2/2-v'0 × t + a' × t2/2

where,

  • x0 is the position of the object in respect to the reference frame at the initial instant,
  • v0 and v0' are the initial velocities of the object and the reference frame respectively,
  • a and a' are the accelerations of the object and the reference frame respectively, and
  • t is the time of motion which is the same for both the object and the reference frame.

We can also use the help of relative kinematic quantities to approach the abovementioned situations. Hence, we can write vrel instead of v - v', v0(rel) instead of v0 - v0' and arel instead of a - a' in all exercises involving the relative motion. In this way, the calculations become easier and shorter.

Relative quantities can be used in other equations of motion with constant acceleration as well. Therefore, we can write

vrel = v0(rel) + arel × t
∆xrel = ( vrel + v0(rel) ) × t/2
v2rel - v20(rel)= 2 × arel × ∆xrel

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  5. Continuing learning kinematics - read our next physics tutorial: Motion. Types of Motion

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