Answer:
The velocity of the ball after the collision is 50 m/s
Explanation:
Hi there!
Since the collision is elastic, the kinetic energy and momentum of the system ball-car is the same before and after the collision. The momentum of the system is the sum of the momentums of each object:
initial momentum = final momentum
mb · vb + mc · vc = mb · vb´ + mc · vc´
Where:
mb = mass of the ball (0.120 kg)
vb = velocity of the ball before the collision (10 m/s).
mc = mass of the car (900 kg).
vc = velocity of the car before the collision (30 m/s).
vb´ = velocity of the ball after the collision.
vc´ = velocity of the car after the collision.
Let´s calculate the initial momentum of the system:
initial momentum = 0.120 kg · 10 m/s + 900 kg · 30 m/s
initial momentum = 27001.2 kg · m/s
Since intial momentum = final momentum
27001.2 kg · m/s = 0.120 kg · vb´ + 900 kg · vc´
The kinetic energy of the system is also conserved:
initial kinetic energy = final kinetic energy
1/2 mb · vb² + 1/2 mc · vc² = 1/2 mb · vb´² + 1/2 mc · vc´²
Let´s calculate the initial kinetic energy:
initial kinetic energy = 1/2 · 0.120 kg · (10 m/s)² + 1/2 · 900 kg · (30 m/s)²
initial kinetic energy = 405006 kg · m²/s². Then:
initial kinetic energy = final kinetic energy
405006 kg · m²/s² = 1/2 · 0.120 kg · vb´² + 1/2 · 900 kg · vc´²
405006 kg · m²/s² = 0.06 kg · vb´² + 450 kg · vc´²
So, we have a system of two equations with two unknowns:
405006 kg · m²/s² = 0.06 kg · vb´² + 450 kg · vc´²
27001.2 kg · m/s = 0.120 kg · vb´ + 900 kg · vc´
Let´s solve the second equation for vc´ (I will omit units for clarity):
27001.2 = 0.120 vb´ + 900 vc´
(27001.2 - 0.120 vb´)/ 900 = vc´
Now let´s replace vc´ in the first equation and solve it for vb´.
405006 = 0.06 vb´² + 450 vc´²
405006 = 0.06 vb´² + 450 [(27001.2 - 0.120 vb´)/ 900]²
405006 = 0.06 vb´² + 450(27001.2² - 6480.288 vb´ + 0.0144 vb´²)/900²
405006 = 0.06 vb´² + 27001.2²/1800 - 3.60016 vb´ + 8 × 10⁻⁶ vb´²
0 = 0.060008 vb´² + 27001.2²/1800 - 405006 - 3.60016 vb´
0 = 0.060008 vb´² - 3.60016 vb´ + 30.0008
Solving the quadratic equation with the quadratic formula:
vb´ = 50 m/s
and
vb´ = 10 m/s
Since 10 m/s is the velocity of the ball before the collision, the solution is vb´ = 50 m/s
