A ball of mass M1 moves with a velocity U1, collides perfectly elastically with a stationary ball of mass M2 . After the collision, M1 and M2 move with velocities V1 and V2 respectively.
Show that : V1 =(M1-M2) / (M1 + M2 ) * U1 .
To show that V1 = (M1 - M2) / (M1 + M2) * U1, we can use the principle of conservation of linear momentum.
Conservation of linear momentum states that the total momentum before the collision is equal to the total momentum after the collision.
Before the collision, the momentum of the first ball (M1) is given by P1 = M1 * U1, where U1 is the initial velocity of the first ball.
After the collision, the momentum of the first ball (M1) is given by P1' = M1 * V1, where V1 is the final velocity of the first ball.
Similarly, the momentum of the second ball (M2) before the collision is zero since it is stationary, and after the collision, the momentum of the second ball (M2) is given by P2' = M2 * V2, where V2 is the final velocity of the second ball.
Since momentum is conserved, we can write:
P1 + P2 = P1' + P2'
M1 * U1 + 0 = M1 * V1 + M2 * V2
Since the collision is perfectly elastic, the relative velocities of the two balls are reversed.
So, V2 - V1 = -U1
Solving the above equation for V1, we get:
V1 = V2 + U1
Plugging this into our momentum conservation equation, we have:
M1 * U1 + 0 = M1 * (V2 + U1) + M2 * V2
Simplifying this equation, we get:
M1 * U1 = M1 * V2 + M1 * U1 + M2 * V2
M1 * U1 - M1 * U1 = M1 * V2 + M2 * V2
0 = V2 * (M1 + M2)
Since V2 cannot be zero (as it's the final velocity of a moving object), we can divide both sides of the equation by (M1 + M2):
0 / (M1 + M2) = V2 * (M1 + M2) / (M1 + M2)
0 = V2
Substituting V2 = 0 in our equation for V1:
V1 = V2 + U1
V1 = 0 + U1
V1 = U1
Therefore, V1 = (M1 - M2) / (M1 + M2) * U1.
This shows that the final velocity of the first ball (V1) is equal to (M1 - M2) / (M1 + M2) times the initial velocity of the first ball (U1).
To show that V1 = (M1 - M2) / (M1 + M2) * U1, let's analyze the conservation of momentum and kinetic energy.
1. Conservation of momentum:
Before the collision, the total momentum is given by:
Initial momentum = M1 * U1 + M2 * 0 (since the second ball is stationary)
After the collision, the total momentum is given by:
Final momentum = M1 * V1 + M2 * V2
According to the principle of conservation of momentum, the initial momentum should be equal to the final momentum:
M1 * U1 + M2 * 0 = M1 * V1 + M2 * V2
Since the second ball is stationary, M2 * 0 = 0, and the equation becomes:
M1 * U1 = M1 * V1 + M2 * V2
2. Conservation of kinetic energy:
For an elastic collision, the total kinetic energy before the collision should be equal to the total kinetic energy after the collision.
Before the collision, the total kinetic energy is given by:
Initial kinetic energy = (1/2) * M1 * U1^2 + (1/2) * M2 * 0^2 (since the second ball is stationary)
After the collision, the total kinetic energy is given by:
Final kinetic energy = (1/2) * M1 * V1^2 + (1/2) * M2 * V2^2
According to the principle of conservation of kinetic energy, the initial kinetic energy should be equal to the final kinetic energy:
(1/2) * M1 * U1^2 + (1/2) * M2 * 0^2 = (1/2) * M1 * V1^2 + (1/2) * M2 * V2^2
Since M2 * 0^2 = 0, the equation becomes:
(1/2) * M1 * U1^2 = (1/2) * M1 * V1^2 + (1/2) * M2 * V2^2
3. Solving the equations:
From step 1, we have: M1 * U1 = M1 * V1 + M2 * V2
Rearranging the equation, we get:
M1 * V1 = M1 * U1 - M2 * V2
Now, we substitute this result into the equation obtained in step 2:
(1/2) * M1 * U1^2 = (1/2) * (M1 * U1 - M2 * V2)^2 + (1/2) * M2 * V2^2
Expanding and simplifying the equation, we get:
M1 * U1^2 = M1^2 * U1^2 + M2^2 * V2^2 - 2 * M1 * M2 * U1 * V2 + M2^2 * V2^2 + M2 * V2^2
Canceling out the common terms and rearranging, we get:
M1 * U1 = M1 * V1 - M2 * V2
Comparing this equation with the one from step 1, we have:
M1 * V1 = M1 * U1 - M2 * V2
Therefore, V1 = (M1 - M2) / M1 * U1
Since M1 + M2 is in the denominator, multiplying both the numerator and denominator by (M1 + M2), we get:
V1 = (M1 - M2) / (M1 + M2) * U1
Hence, we have successfully shown that V1 = (M1 - M2) / (M1 + M2) * U1.