A ball A of mass M collides elastically with another identical ball B at rest .Initially velocity of ball A is u m/s.After collision
To find the final velocities of both balls A and B after the elastic collision, we can use the principles of conservation of momentum and kinetic energy.
1. Conservation of momentum:
In an isolated system, the total momentum before the collision is equal to the total momentum after the collision. Since ball B is initially at rest, the momentum of ball A before the collision is given by:
Momentum(A) = mass(A) * velocity(A) = M * u
After the collision, let the final velocities of balls A and B be v1 and v2 respectively. The momentum of ball A after the collision is M * v1, and ball B remains at rest, so the momentum of ball B after the collision is 0.
Using the principle of conservation of momentum:
Momentum(before) = Momentum(after)
M * u = M * v1 + 0
M * u = M * v1
v1 = u
Therefore, ball A continues to move in the same direction with the same speed (u) after the collision.
2. Conservation of kinetic energy:
In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision. The kinetic energy of a moving object is given by:
Kinetic energy = (1/2) * mass * velocity^2
The kinetic energy before the collision is:
KE(before) = (1/2) * M * u^2
The kinetic energy after the collision is the sum of the kinetic energies of ball A and B:
KE(after) = (1/2) * M * v1^2 + (1/2) * M * v2^2
Using the principle of conservation of kinetic energy:
KE(before) = KE(after)
(1/2) * M * u^2 = (1/2) * M * v1^2 + (1/2) * M * v2^2
We know that v1 = u, so substituting and simplifying:
(1/2) * M * u^2 = (1/2) * M * u^2 + (1/2) * M * v2^2
0 = (1/2) * M * v2^2
From this, we can conclude that the final velocity of ball B after the collision is 0 m/s. Ball B comes to rest due to the collision.
To summarize:
- The final velocity of ball A after the collision remains unchanged and is equal to its initial velocity (u) in the same direction.
- The final velocity of ball B after the collision is 0 m/s. It comes to rest.
Note: This explanation assumes an idealized scenario of an elastic collision, where no external forces are acting and there is no loss of kinetic energy. In real-world situations, some energy may be lost due to factors like friction or other forms of energy dissipation.