A rubber ball of mass m1= 10kg is moving to the right with speed v1= 10m/s. it collides elastically with another ball of mass m2= 50kg, which is sitting at rest. m2 is larger than m1, what are the speeds of the balls, v1 and v2 after the collision?
To determine the speeds of the balls after the collision, we can use the principles of conservation of momentum and kinetic energy.
1. Conservation of Momentum:
According to the law of conservation of momentum, the total momentum of a system remains constant before and after a collision, provided no external forces act on the system. Mathematically, we can write:
m1 * v1(initial) + m2 * v2(initial) = m1 * v1(final) + m2 * v2(final)
Since the second ball is initially at rest (v2(initial) = 0), this equation simplifies to:
m1 * v1(initial) = m1 * v1(final) + m2 * v2(final)
Therefore, the momentum before the collision is equal to the momentum after the collision.
2. Conservation of Kinetic Energy:
In an elastic collision, the total kinetic energy of the system is conserved. Mathematically, we can write:
(1/2) * m1 * v1(initial)^2 + (1/2) * m2 * v2(initial)^2 = (1/2) * m1 * v1(final)^2 + (1/2) * m2 * v2(final)^2
Since the second ball is initially at rest (v2(initial) = 0), this equation simplifies to:
(1/2) * m1 * v1(initial)^2 = (1/2) * m1 * v1(final)^2 + (1/2) * m2 * v2(final)^2
Therefore, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.
Using the above equations, we can solve for v1(final) and v2(final).
Given:
m1 = 10 kg (mass of the first ball)
v1(initial) = 10 m/s (initial speed of the first ball)
m2 = 50 kg (mass of the second ball)
v2(initial) = 0 m/s (initial speed of the second ball)
Substituting the given values into the equations:
10 kg * 10 m/s = 10 kg * v1(final) + 50 kg * v2(final) (1) (from conservation of momentum)
0.5 * 10 kg * (10 m/s)^2 = 0.5 * 10 kg * v1(final)^2 + 0.5 * 50 kg * v2(final)^2 (2) (from conservation of kinetic energy)
Now, we can solve these equations simultaneously to find v1(final) and v2(final).