In one dimension, a 12 kg ball moving to the right at 9 m/s collides with a 4kg ball moving to the left at 3 m/s. The collision is perfectly elastic. Find the final velocities of both balls.
To find the final velocities of both balls after the collision, we can apply the principles of conservation of momentum and kinetic energy.
First, let's calculate the initial momentum of each ball. Momentum is defined as the product of an object's mass and velocity.
For the 12 kg ball moving to the right:
Initial momentum = mass × velocity = 12 kg × 9 m/s = 108 kg⋅m/s (to the right)
For the 4 kg ball moving to the left:
Initial momentum = mass × velocity = 4 kg × (-3 m/s) = -12 kg⋅m/s (to the left)
Since momentum is conserved in an elastic collision, the total initial momentum equals the total final momentum.
Therefore, the total initial momentum = total final momentum
108 kg⋅m/s + (-12 kg⋅m/s) = (12 kg × final velocity₁) + (4 kg × final velocity₂)
Simplifying the equation:
96 kg⋅m/s = 12 kg⋅final velocity₁ + 4 kg⋅final velocity₂
Now, let's apply the principle of conservation of kinetic energy. In an elastic collision, the total kinetic energy of the system remains constant.
The total initial kinetic energy = the total final kinetic energy
Calculating the initial kinetic energy of the 12 kg ball:
Initial kinetic energy = (1/2) × mass × (velocity)²
= (1/2) × 12 kg × (9 m/s)²
= 486 J
Calculating the initial kinetic energy of the 4 kg ball:
Initial kinetic energy = (1/2) × mass × (velocity)²
= (1/2) × 4 kg × (-3 m/s)²
= 18 J
Therefore, the total initial kinetic energy = 486 J + 18 J = 504 J
Now, let's calculate the final kinetic energy using the final velocities:
Calculating the final kinetic energy of the 12 kg ball:
Final kinetic energy₁ = (1/2) × 12 kg × (final velocity₁)²
Calculating the final kinetic energy of the 4 kg ball:
Final kinetic energy₂ = (1/2) × 4 kg × (final velocity₂)²
Setting the total final kinetic energy equal to the initial kinetic energy:
504 J = Final kinetic energy₁ + Final kinetic energy₂
504 J = (1/2) × 12 kg × (final velocity₁)² + (1/2) × 4 kg × (final velocity₂)²
Now, we have two equations:
1. 96 kg⋅m/s = 12 kg⋅final velocity₁ + 4 kg⋅final velocity₂
2. 504 J = (1/2) × 12 kg × (final velocity₁)² + (1/2) × 4 kg × (final velocity₂)²
We can solve these two equations simultaneously to find the final velocities of both balls.