That the total momentum is the same before and after a collision is a basic rule of mechanics. By the principle of relativity, if it holds in one frame it must hold in every frame.
But with momentum taken as , a sum that is conserved in one frame stops balancing before and after once the velocities are translated into another frame with the addition rule. The figure shows one collision from two frames, and in the lower row the sums before and after differ.
Since velocities no longer combine by plain addition, the formula for momentum has to be rewritten too.
The denominator of the velocity is time measured on the observer's clock, and it changes from frame to frame.
Divide instead by the proper time of a clock moving with the body. From , , and momentum takes this form:
pairs with the energy to form the four-momentum. If the sum of four-momenta is conserved in one frame it is conserved in all, so is conserved in every frame.
Keep applying a force and the momentum grows without limit. The speed only approaches without passing it; the growth is taken up by .
At , ; at , ; at , . The speed barely changes while the momentum changes a great deal.
Protons accelerated at the LHC have an energy of TeV, of about , and a speed of percent of the speed of light. What the acceleration increases is almost entirely momentum and energy.
is sometimes read as , with the explanation that a faster body has more mass. This way of speaking is no longer used.
Depending on whether a force is applied along the motion or across it, the resistance to acceleration is or , which no single increased mass can describe.
The mass is one number, measured in the frame where the body is at rest, and it is the same in every frame. The ball in the figure moves at , , and , with equal to , , and , but its mass does not change.
When is much smaller than , and .
The two curves in the figure almost coincide at , where exceeds by only percent. At it is times larger.
The rewritten momentum contains Newtonian as its approximation at low speed.