# Relative Motion

## Relative motion

All motion is relative to some frame of reference. The simplest laboratory frame of reference is three mutually perpendicular axes at rest with respect to an observer. In terms of the frame of reference of an observer some distance from Earth, the laboratory frame of reference would be moving with Earth as it rotates on its axis and as it revolves about the Sun. What would be a simple form of motion in the laboratory frame of reference would appear to be a much more complicated motion in the frame of reference of the distant observer. *See* Frame of reference

Motion means continuous change of position of an object with respect to an observer. To another observer in a different frame of reference the object may not be moving at all, or it may be moving in an entirely different manner. The motions of the planets were found in ancient times to appear quite complicated in the laboratory frame of reference of an observer on Earth. By transferring to the frame of reference of an imaginary observer on the Sun, Johannes Kepler showed that the relative motion of the planets could be simply described in terms of elliptical orbits. The validity of one description is no greater than the other, but the latter description is far more convenient.

## Relative Motion

the motion of a point or a body with respect to a moving frame of reference that travels in a certain manner relative to some other, primary frame of reference arbitrarily called fixed. The velocity of a point in relative motion is called the relative velocity v_{rel}, and the point’s acceleration is referred to as the relative acceleration w_{rel}. The motion of all points of the moving frame of reference with respect to the fixed frame is in this case called vehicle motion, and the velocity and acceleration of the point of the moving system through which the point in motion passes at a given moment of time are called the vehicle velocity v_{veh} and the vehicle acceleration w_{veh}, respectively.

The motion of a point or a body with respect to a fixed frame of reference is said to be compound or absolute, and the velocity and acceleration of this motion are called the absolute velocity V_{a} and absolute acceleration w_{a}, respectively. For example, let us define a steamship as a moving frame of reference and the shore as the fixed frame of reference. The motion of a ball rolling on the deck of the steamship will then be relative motion with respect to the deck and absolute motion with respect to the shore. The velocity and acceleration of the ball will be v_{rel} and w_{rel}, respectively, in the first case and v_{a} and w_{a}, respectively, in the second case. The motion of the entire steamship with respect to the shore will be vehicle motion for the ball. If the ball is considered as a point, the velocity and acceleration of the point on the deck touched by the ball at a given moment will be v_{veh} and w_{veh}. The relation between these quantities is given in classical mechanics by the eauations

(1) *V _{u} = V_{rel} + V_{reh} W_{a} = W_{rel} + W_{veh} + W_{Cor}*

where W_{cor} is the Coriolis acceleration. The formulas of (1) are widely used in kinematics in the study of the motion of points and bodies.

In dynamics, motion with respect to a noninertial frame of reference, for which Newton’s law of mechanics are invalid, is called relative motion. In order for the equations of relative motion of a mass point to retain the same form as in an inertial frame of reference, it is necessary to add a vehicle force of inertia J_{veh} = — mw_{Veh} and the Coriolis force of inertia J_{cor} = — mw_{cor} to the force F that acts on the point, where *m* is the mass of the point and F is the force of interaction with other bodies. Then

(2) *m* W_{rel} = F + J_{veh} + J_{Cor}

For the relative motion of a system of mass points, similar equations are constructed for all points of the system. These equations are used to study the relative motion of various mechanical devices, particularly gyroscopes, that are mounted on moving foundations such as ships, aircraft, and rockets. The equations are also used in studying the motion of bodies with respect to the earth in cases where the earth’s diurnal rotation must be taken into consideration.

S. M. TARG