A high-speed photograph of a club hitting a golf ball is shown in the figure below. The club was in contact with a ball, initially at rest, for about 0.0037 s. If the ball has a mass of 55 g and leaves the head of the club with a speed of 2.7 102 ft/s, find the average force exerted on the ball by the club.
Answer should be in kN
To find the average force exerted on the ball by the club, we can use the impulse-momentum principle, which states that the change in momentum of an object is equal to the net force applied to it multiplied by the time interval over which the force is applied.
First, we need to find the initial momentum of the ball. The initial momentum (p_i) can be calculated using the formula:
p_i = m * v_i,
where m is the mass of the ball and v_i is the initial velocity of the ball.
Given that the mass of the ball (m) is 55 g, we need to convert it to kilograms by dividing it by 1000:
m = 55 g / 1000 = 0.055 kg.
Given that the initial velocity (v_i) is 0 ft/s since the ball is initially at rest, the initial momentum (p_i) is:
p_i = 0.055 kg * 0 ft/s = 0 kg·m/s.
Next, we need to find the final momentum of the ball. The final momentum (p_f) can be calculated using the formula:
p_f = m * v_f,
where v_f is the final velocity of the ball.
Given that the final velocity (v_f) is 2.7 * 10^2 ft/s, we need to convert it to meters per second (m/s) by multiplying it by a conversion factor of 0.3048 m/ft:
v_f = 2.7 * 10^2 ft/s * 0.3048 m/ft = 82.3 m/s.
Now, we can calculate the final momentum (p_f) using the mass and final velocity:
p_f = 0.055 kg * 82.3 m/s = 4.53 kg·m/s.
Since impulse (J) is equal to the change in momentum, we can calculate it using the formula:
J = p_f - p_i.
J = 4.53 kg·m/s - 0 kg·m/s = 4.53 kg·m/s.
Finally, the average force (F) exerted on the ball by the club can be calculated by dividing the impulse by the time duration (t) over which the force is applied:
F = J / t.
Given that the time duration (t) is 0.0037 s, we can calculate the average force (F):
F = 4.53 kg·m/s / 0.0037 s = 1224.32 N.
However, the answer should be in kilonewtons (kN). To convert from newtons (N) to kilonewtons (kN), we divide the value by 1000:
F = 1224.32 N / 1000 = 1.22432 kN.
Therefore, the average force exerted on the ball by the club is approximately 1.22432 kN.
To find the average force exerted on the ball by the club, we can use the impulse-momentum principle.
The impulse-momentum principle states that the change in momentum of an object is equal to the force applied to it multiplied by the time interval over which the force is applied. Mathematically, it can be expressed as:
Impulse = Force × Time
We can rearrange this equation to solve for force:
Force = Impulse / Time
The impulse can be calculated by finding the change in momentum of the ball. The initial momentum of the ball is zero since it is at rest, and the final momentum can be calculated using the equation:
Momentum = Mass × Velocity
So, the change in momentum can be written as:
Δp = Mass × Final Velocity - Mass × Initial Velocity
Let's plug in the values given in the problem:
Mass = 55 g = 0.055 kg
Final Velocity = 2.7 × 10^2 ft/s = 82.3 m/s
Initial Velocity = 0 ft/s (since the ball is at rest)
Now, let's calculate the change in momentum:
Δp = (0.055 kg) × (82.3 m/s - 0 m/s)
= 4.5265 kg·m/s
Next, we can determine the average force by dividing the change in momentum by the time:
Time = 0.0037 s
Force = (4.5265 kg·m/s) / (0.0037 s)
= 1225.81 N
Finally, let's convert the force to kN:
Force = 1225.81 N / 1000
= 1.22581 kN
Therefore, the average force exerted on the ball by the club is approximately 1.22581 kN.