Ball 2, with a mass of 30kg, is moving to the left at 10m/s. With what velocity should Ball 1, with a mass of 10kg, move to the right and collide with Ball 2, so that Ball 1 rebounds with a velocity of 30m/s, and Ball 2 with a velocity of 10 m/s, after the collision? Assume the collision to be perfectly elastic.
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To find the velocity at which Ball 1 should move to the right in order to achieve the desired outcome, we can apply the principles of conservation of momentum and kinetic energy.
Conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. Mathematically, this can be expressed as:
m1 * v1_initial + m2 * v2_initial = m1 * v1_final + m2 * v2_final
where:
m1 and m2 are the masses of Ball 1 and Ball 2, respectively,
v1_initial and v2_initial are the initial velocities of Ball 1 and Ball 2, and
v1_final and v2_final are the final velocities of Ball 1 and Ball 2.
In this case, we have:
m1 = 10 kg, v1_initial = ? (what we need to find),
m2 = 30 kg, v2_initial = -10 m/s (negative because Ball 2 is moving to the left),
v1_final = 30 m/s (given),
v2_final = 10 m/s (given).
Now, let's plug the given values into the momentum conservation equation:
(10 kg)(v1_initial) + (30 kg)(-10 m/s) = (10 kg)(30 m/s) + (30 kg)(10 m/s)
Simplifying this equation will give us the value of v1_initial:
10v1_initial - 300 = 300 + 300
10v1_initial = 600 + 300
10v1_initial = 900
v1_initial = 900 / 10
v1_initial = 90 m/s
Therefore, in order for Ball 1 to rebound with a velocity of 30 m/s after the collision, Ball 1 should move to the right with a velocity of 90 m/s before the collision occurs.