Suppose that high-speed stroboscopic photographs show that the 200-g head of a golf club travels at 51 m/s just before it strikes a 50-g golf ball initially at rest on a tee.
If the club head slows to at 36 m/s (in the same direction) after the collision,
what is the speed of the golf ball just after impact, and
what fraction of the original kinetic energy of the club head was lost to thermal energy? Assume that the mass of the club is concentrated mostly in the head.
The loss of momentum by the clud head equals the momentum acquired by the gold ball. Use that and its mass to derive the speed of the golf ball.
Then add up the kinetic energies before the collision (club head only) and afterwards (club head plus ball). Divide the before/after difference by the initial club head KE to get the fractional energy loss.
To find the speed of the golf ball just after impact, we can use the principle of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.
Momentum before collision = Momentum after collision
The momentum of an object is given by the product of its mass and velocity. Let's denote the velocity of the golf ball after impact as v1 and the velocity of the golf club head after impact as v2.
The total momentum before the collision is the sum of the momentum of the golf ball and the golf club head:
200 g (0.051 m/s) + 50 g (0 m/s) = (200 g + 50 g) v1
Converting the masses to kilograms:
(0.2 kg)(0.051 m/s) + (0.05 kg)(0 m/s) = (0.25 kg) v1
v1 = (0.2 kg)(0.051 m/s) / (0.25 kg)
v1 ≈ 0.0408 m/s
So, the speed of the golf ball just after impact is approximately 0.0408 m/s.
To calculate the fraction of the original kinetic energy of the club head lost to thermal energy, we need to find the initial and final kinetic energies of the club head. The kinetic energy of an object is given by the equation:
Kinetic Energy = (1/2) * mass * velocity^2
Let's denote the initial velocity of the club head as v0. The initial kinetic energy of the club head is:
Initial Kinetic Energy = (1/2) * (0.2 kg) * (51 m/s)^2
And the final kinetic energy of the club head is:
Final Kinetic Energy = (1/2) * (0.2 kg) * (36 m/s)^2
The fraction of kinetic energy lost to thermal energy can be calculated using the formula:
Fraction of energy lost = (Initial Kinetic Energy - Final Kinetic Energy) / Initial Kinetic Energy
Plugging in the values:
Fraction of energy lost = [(1/2) * (0.2 kg) * (51 m/s)^2 - (1/2) * (0.2 kg) * (36 m/s)^2] / [(1/2) * (0.2 kg) * (51 m/s)^2]
Simplifying the equation will give you the fraction of the original kinetic energy lost to thermal energy.