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.
10.5
To solve this problem, we can use the principles of conservation of momentum and conservation of energy.
Step 1: Find the velocity of the golf ball after impact.
Using the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision. In this case, the only objects involved are the golf club head and the golf ball.
So, we can write the equation as:
(mass of club head * velocity of club head) + (mass of golf ball * velocity of golf ball) = (mass of club head * final velocity of club head) + (mass of golf ball * final velocity of golf ball)
Let's substitute the given values:
(0.200 kg * 51 m/s) + (0.050 kg * 0 m/s) = (0.200 kg * 36 m/s) + (0.050 kg * final velocity of golf ball)
Simplifying this equation will give us the final velocity of the golf ball.
Step 2: Find the fraction of kinetic energy lost to thermal energy.
The initial kinetic energy of the club head is given by:
(1/2) * (mass of club head) * (velocity of club head)^2
The final kinetic energy of the club head is given by:
(1/2) * (mass of club head) * (final velocity of club head)^2
The fraction of kinetic energy lost to thermal energy is the difference between the initial and final kinetic energies, divided by the initial kinetic energy.
Let's calculate both the speed of the golf ball after impact and the fraction of kinetic energy lost to thermal energy:
Step 1:
(0.200 kg * 51 m/s) + (0.050 kg * 0 m/s) = (0.200 kg * 36 m/s) + (0.050 kg * final velocity of golf ball)
10.2 kg·m/s = 7.2 kg·m/s + 0.050 kg·final velocity of golf ball
Subtracting 7.2 kg·m/s from both sides:
3 kg·m/s = 0.050 kg·final velocity of golf ball
Dividing both sides by 0.050 kg:
final velocity of golf ball = 60 m/s
Step 2:
Initial kinetic energy of the club head = (1/2) * (mass of club head) * (velocity of club head)^2
Initial kinetic energy = (1/2) * (0.200 kg) * (51 m/s)^2
Final kinetic energy of the club head = (1/2) * (mass of club head) * (final velocity of club head)^2
Final kinetic energy = (1/2) * (0.200 kg) * (36 m/s)^2
Kinetic energy lost to thermal energy = Initial kinetic energy - Final kinetic energy
Fraction of kinetic energy lost to thermal energy = (Kinetic energy lost to thermal energy) / (Initial kinetic energy)
Let's calculate the fraction of kinetic energy lost to thermal energy:
Initial kinetic energy = (1/2) * (0.200 kg) * (51 m/s)^2
Final kinetic energy = (1/2) * (0.200 kg) * (36 m/s)^2
Kinetic energy lost to thermal energy = (1/2) * (0.200 kg) * (51 m/s)^2 - (1/2) * (0.200 kg) * (36 m/s)^2
Fraction of kinetic energy lost to thermal energy = (Kinetic energy lost to thermal energy) / (Initial kinetic energy)