A golf ball of mass 125 g hits a billiards ball of mass 172.5 g that is initially at rest. After the collision, the golf ball moves to the right at a speed of 10 cm/s. The billiards ball moves to the left with a speed of 58 cm/s. (a) What was the velocity of the golf ball before the collision? (b) What is the change in total kinetic energy of the system at the collision?
m1v1i = miv1f + m2v2f
solve for v1i (watch your directions!)
part b is just the difference of KEi for the one ball and KEf for both balls
To solve this problem, we can apply the principle of conservation of momentum and the equation for kinetic energy. Now let's solve each part of the problem step by step.
(a) What was the velocity of the golf ball before the collision?
The principle of 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:
m₁v₁ + m₂v₂ = m₁v₁' + m₂v₂'
Where:
m₁ and m₂ are the masses of the golf ball and billiards ball respectively,
v₁ and v₂ are the initial velocities of the golf ball and billiards ball respectively,
v₁' and v₂' are the final velocities of the golf ball and billiards ball respectively.
We know the masses of the golf ball and billiards ball:
m₁ = 125 g = 0.125 kg
m₂ = 172.5 g = 0.1725 kg
We also know the final velocities of the golf ball and billiards ball:
v₁' = 10 cm/s = 0.1 m/s (because the golf ball moves to the right)
v₂' = -58 cm/s = -0.58 m/s (because the billiards ball moves to the left)
Since the billiards ball is initially at rest (v₂ = 0), we can rewrite the equation as:
m₁v₁ = m₁v₁' + m₂v₂'
Substituting the known values, we have:
0.125 kg * v₁ = 0.125 kg * 0.1 m/s + 0.1725 kg * (-0.58 m/s)
Solving for v₁:
v₁ = (0.125 kg * 0.1 m/s + 0.1725 kg * (-0.58 m/s)) / 0.125 kg
v₁ ≈ -0.46 m/s
Therefore, the velocity of the golf ball before the collision is approximately -0.46 m/s to the left.
(b) What is the change in total kinetic energy of the system at the collision?
The change in total kinetic energy of the system can be calculated by subtracting the initial kinetic energy from the final kinetic energy.
The initial kinetic energy is given by:
K₁ = (1/2) * m₁ * v₁² + (1/2) * m₂ * v₂²
Substituting the known values:
K₁ = (1/2) * 0.125 kg * (-0.46 m/s)² + (1/2) * 0.1725 kg * 0 m/s²
K₁ ≈ 0.011 kg * m²/s²
The final kinetic energy is given by:
K₂ = (1/2) * m₁ * v₁'² + (1/2) * m₂ * v₂'²
Substituting the known values:
K₂ = (1/2) * 0.125 kg * 0.1 m/s² + (1/2) * 0.1725 kg * (-0.58 m/s)²
K₂ ≈ 0.003125 kg * m²/s²
The change in total kinetic energy can be calculated as:
ΔK = K₂ - K₁
ΔK ≈ 0.003125 kg * m²/s² - 0.011 kg * m²/s²
ΔK ≈ -0.007875 kg * m²/s²
Therefore, the change in total kinetic energy of the system at the collision is approximately -0.007875 kg * m²/s².