two perfect elastic balls of same mass m are moving with velocities u1 and u2 . they collide elastically n times. the kinetic energy of the system finally is?
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To determine the kinetic energy of the system after n elastic collisions, we need to consider the conservation of kinetic energy and momentum during each collision.
When two objects collide elastically, the total kinetic energy before the collision is equal to the total kinetic energy after the collision. Additionally, the total momentum of the system (the sum of the momenta of the two objects) is conserved.
Let's analyze the collisions step by step:
1. Initial collision:
- The first ball of mass m with velocity u1 collides with the second ball of mass m with velocity u2.
- Conservation of momentum:
- Before the collision: m * u1 + m * u2
- After the collision: m * v1 + m * v2 (where v1 and v2 are the velocities of the balls after the collision)
- As the collision is elastic, momentum is conserved: m * u1 + m * u2 = m * v1 + m * v2
2. Subsequent collisions:
- After the initial collision, the first ball collides with the second ball n-1 more times.
- Each collision preserves momentum and kinetic energy.
Given the collisions are elastic, we can conclude:
- Momentum is conserved throughout every collision.
- The kinetic energy of the system is constant before and after each collision.
Therefore, the kinetic energy of the system after n elastic collisions is the same as the initial kinetic energy of the system.
The initial kinetic energy of the system is given by:
Kinetic energy = (1/2) * m * u1^2 + (1/2) * m * u2^2
Hence, the kinetic energy of the system after n elastic collisions is (1/2) * m * u1^2 + (1/2) * m * u2^2.