A New Look at Momentum
Suppose that a number of observers, each in a different inertial reference frame, watch an isolated collision between two particles. In classical mechanics, we have seen that-even though the observers measure different velocities for the colliding particles-they all find that the law of conservation of momentum holds. That is.they find that the total momentum of the system of particles after the collision is the same as it was before the collision.How is this situation affected by relativity? We find that if we continue to define the momentum p of a panicle as ml the product of its mass and its velocity, total momentum is not conserved for the observers in different inertial frames. We have two choices: (I) Give up the law of conservation of momentum or (2) see if we can redefine the momentum of a particle in some new way so that the law of conservation of momentum still holds. The correct choice is the second one.Consider a particle moving with constant speed v in the positive R.direction. Classically, its momentum has magnitude
in which I1x is the distance it travels in time I1t. To find a relativistic expression for momentum, we start with the new definition.
Here, as before, Ax is the distance traveled by a moving particle as viewed by an observer watching that particle. However, I1to is the time required to travel that distance, measured not by the observer watching the moving particle but ‘by an observer moving with the particle. The particle is at rest with respect to this second0 observer, with the result that the time this observer measures is a proper time. Using the time dilation formula (Eq, 38·9), we can then write .
Note that this differs from the classical definition of Eq. 38-37 only by the Lorentz factor y. However, that difference is important: Unlike classical momentum, relativistic momentum approaches an infinite value as v approaches c.
We can generalize the definition of Eq. 38-38 to vector form as
This equation gives the correct definition of momentum for all physically possible speeds. For a speed much less than c, it reduces to the c1assl~al definition of momentum <p = mv).