Momentum Conservation Mass Variation

Momentum and energy conservation laws are the consequence of homogeneity of space and time respectively. We therefore assume that these laws are valid also in special theory of relativity where particles, or frames of references, are moving with velocities comparable to that of light. In fact, the inter-relationship between space and time co-ordinates in special relativity implies that if momentum is conserved, energy is automatically conserved.

Assuming conservation of linear momentum, let us consider a perfectly elastic collision between two identical spheres, A and B. Further, we view this process from Lorentz frames S and S’, S’ moving with velocity v relative to S along common X, X’-axis. To begin with, let the sphere A be at rest in frame S and B at rest in frame S’.

Now suppose that the observer in S projects sphere A along Y-axis with velocity U₀ in his frame, and observer in S’ projects sphere B along Y’-axis with velocity –U₀ in his frame. The spheres are projected in such a way that they collide head-on; at the moment of collision, centers of both A and B lie along Y (or Y’) axis.

Let us first view the process from frame S. Before collision, the velocities of A and B are:

(U_A)_x = 0,                   (U_A)_y = U₀



where, = (1 – v²/c²)–1/2. We have used equation to determine components of velocity of sphere B in frame S; in frame S’, (U_B)_x’ = 0 and (U_B)_y’ = –U₀

During collision, transverse component of velocities do not change, i.e. X-component of velocity of sphere A remains zero and that of B remains equal to v. The Y-components reverse their directions, since spheres are identical. That is, after collision, sphere A moves with velocity –U₀ and B moves with velocity U₀/ along Y-axis, as seen in frame S. The process in frame S looks as shown in fig.

It is trivial to see that momentum remains conserved about X-axis. The conservation of momentum along Y-axis, as observed in frame S, implies



As observed from frame S’, the roles of A and B are interchanged and the sign of v is reversed. Now, sphere A is moving with velocity –v along X’-axis while B has no velocity along X’-axis. Following same analysis as above, in frame S’, we get

m_A = m_B

The process as its looks from S’ is shown in fig.

Thus, we find that conservation of momentum implies that mass of a body becomes different when viewed from different frames.

In the limit sphere A is at rest in frame S and sphere B moves along X-axis with velocity v; B is otherwise identical to A. If we put m_A = m₀ as the mass of sphere A at rest, then mass of moving sphere (as seen from S) is given by



The converse situation holds in frame S’. In the limit m_B = m₀ and m_A is mass of sphere A with speed v along negative X’-axis. Once again, it leads to the above relation.

Thus, special relativity shows that mass of body depends upon its speed; as speed increases, mass also increases, as following:



If v << c, we find m (v) ≃ m₀, where m₀ is at rest mass of the body.

The relativistic momentum of a material body is therefore defined as,

Services: - Momentum Conservation, Mass Variation Homework | Momentum Conservation, Mass Variation Homework Help | Momentum Conservation, Mass Variation Homework Help Services | Live Momentum Conservation, Mass Variation Homework Help | Momentum Conservation, Mass Variation Homework Tutors | Online Momentum Conservation, Mass Variation Homework Help | Momentum Conservation, Mass Variation Tutors | Online Momentum Conservation, Mass Variation Tutors | Momentum Conservation, Mass Variation Homework Services | Momentum Conservation, Mass Variation