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Dependence on reference frame

Momentum is a measurable quantity, and the measurement depends on the frame of reference. For example: if an aircraft of mass 1000 kg is flying through the air at a speed of 50 m/s its momentum can be calculated to be 50,000 kg.m/s. If the aircraft is flying into a headwind of 5 m/s its speed relative to the surface of the Earth is only 45 m/s and its momentum can be calculated to be 45,000 kg.m/s. Both calculations are equally correct. In both frames of reference, any change in momentum will be found to be consistent with the relevant laws of physics.

Suppose x is a position in an inertial frame of reference. From the point of view of another frame of reference, moving at a constant speed u relative to the other, the position (represented by a primed coordinate) changes with time as

x ′ = x − u t. {\displaystyle x'=x-ut\,.}

This is called a Galilean transformation.

If a particle is moving at speed d x / d t = v in the first frame of reference, in the second, it is moving at speed

v ′ = d x ′ d t = v − u. {\displaystyle v'={\frac {{\text{d}}x'}{{\text{d}}t}}=v-u\,.}

Since u does not change, the second reference frame is also an inertial frame and the accelerations are the same:

a ′ = d v ′ d t = a. {\displaystyle a'={\frac {{\text{d}}v'}{{\text{d}}t}}=a\,.}

Thus, momentum is conserved in both reference frames. Moreover, as long as the force has the same form, in both frames, Newton's second law is unchanged. Forces such as Newtonian gravity, which depend only on the scalar distance between objects, satisfy this criterion. This independence of reference frame is called Newtonian relativity or Galilean invariance.

A change of reference frame, can, often, simplify calculations of motion. For example, in a collision of two particles, a reference frame can be chosen, where, one particle begins at rest. Another, commonly used reference frame, is the center of mass frame – one that is moving with the center of mass. In this frame, the total momentum is zero.