A 5.00 kg ball (ball 1), moving to the right at a velocity of +4.00 m/s on a frictionless table, collides head-on with a stationary 7.60 kg ball (ball 2). Find the final velocities of the balls if the collision is as specified below.
(a) elastic collision
ball 1
ball 2
(b) completely inelastic collision
ball 1
ball 2
To calculate the final velocities of the balls in each scenario, we can use the principle of conservation of momentum.
(a) Elastic Collision:
In an elastic collision, both momentum and kinetic energy are conserved.
Momentum conservation: The total momentum before the collision must be equal to the total momentum after the collision.
The initial momentum of ball 1 is given by p₁ = m₁ * v₁ = (5.00 kg) * (+4.00 m/s) = +20.00 kg·m/s.
The initial momentum of ball 2 is zero since it is stationary, p₂ = 0 kg·m/s.
The final velocities of the balls can be represented by v₁f (final velocity of ball 1) and v₂f (final velocity of ball 2).
Using momentum conservation, we have:
p₁ + p₂ = m₁ * v₁f + m₂ * v₂f
Since the final momentum of ball 2 is zero (it remains stationary),
p₁ = m₁ * v₁f
20.00 kg·m/s = (5.00 kg) * v₁f
v₁f = 20.00 kg·m/s / 5.00 kg
v₁f = +4.00 m/s
So the final velocity of ball 1 is +4.00 m/s.
To find the final velocity of ball 2, we can use the equation:
p₁ + p₂ = m₁ * v₁f + m₂ * v₂f
20.00 kg·m/s + 0 kg·m/s = (5.00 kg) * (+4.00 m/s) + (7.60 kg) * v₂f
We can now solve for v₂f:
v₂f = (20.00 kg·m/s - (5.00 kg) * (+4.00 m/s)) / (7.60 kg)
v₂f = -12.63 m/s
So the final velocity of ball 2 is -12.63 m/s in the opposite direction to the initial velocity of ball 1.
(b) Completely Inelastic Collision:
In a completely inelastic collision, the two balls stick together after the collision and move as a single object.
Again, using momentum conservation, we have:
p₁ + p₂ = (m₁ + m₂) * vf
Since the balls stick together, the final velocity of the combined balls is the same.
p₁ + p₂ = (5.00 kg) * vf + (7.60 kg) * vf
Using the previous values for p₁ and p₂:
20.00 kg·m/s + 0 kg·m/s = (12.60 kg) * vf
We can now solve for vf (final velocity of the combined balls):
vf = (20.00 kg·m/s) / (12.60 kg)
vf = +1.59 m/s
So the final velocities of the balls in a completely inelastic collision are both +1.59 m/s in the same direction.