Question

Derive the formula flr the ginal velocities V1 and V2 in an isolated system consist of two non rotating spheres having masses m and 2m sliding in one dimension with initial velocities U1 and U2 (U1>U2) collides elastically

Answer 1

The final velocities in an elastic collision involving two spheres of masses m and 2m can be derived using conservation of momentum and energy laws. By solving the system of equations, we obtain V1 = U1/3 + 4U2/3 and V2 = 2U1/3 + U2/3.

In an elastic collision of two non-rotating spheres with masses m and 2m sliding in one dimension, we can use conservation laws to derive the formulas for the final velocities V1 and V2.

1. Momentum Conservation: The total momentum before the collision equals the total momentum after the collision. This can be expressed as:
   mU1 + 2mU2 = mV1 + 2mV2
2. Kinetic Energy Conservation: Since the collision is elastic, the total kinetic energy before the collision equals the total kinetic energy after the collision:
   (1/2)mU1² + (1/2)2mU2² = (1/2)mV1² + (1/2)2mV2²
3. Solving the System of Equations: We have two equations:
   1. mU1 + 2mU2 = mV1 + 2mV2
   2. U1² + 2U2² = V1² + 2V2²

Rearrange and solve these equations to find:

Final Velocities:

V1 = (U1*(m - 2m) + 2*2m*U2) / (m + 2m) = (U1/3) + (4U2/3)

V2 = (U1*(m) + U2*(2m - m)) / (m + 2m) = (2U1/3) + (U2/3)