Question

A golfer hits a golf ball, imparting to it an initial velocity of magnitude \(52.2 \mathrm{~m} / \mathrm{s}\) directed \(30^{\circ}\) above the horizontal. Assuming that the mass of the ball is \(46.0 \mathrm{~g}\) and the club and ball are in contact for \(1.20 \mathrm{~ms}\), find \((a)\) the impulse imparted to the ball, \((b)\) the impulse imparted to the club, and \((c)\) the average force exerted on the ball by the club.

Short answer

The impulse imparted to the ball is \(23.94 ~kg·m/s\) directed \(30°\) above the horizontal. The impulse imparted to the club is the same magnitude, \(23.94 ~kg·m/s\) but directed \(30°\) below the horizontal. The average force exerted on the ball by the club is \(19950 ~N\) directed \(30°\) above the horizontal.

Step-by-step solution

Calculate Final Velocity

We start by calculating the final velocity of the golf ball after impact with the club, using the initial velocity and the angle. Since the velocity is at an angle, it has both horizontal and vertical components. We use trigonometry to find these components, \(v_{fx} = 52.2 \cos(30)\) for the horizontal component and \(v_{fy} = 52.2 \sin(30)\) for the vertical component.

Find Impulse Imparted to the Ball

Next we calculate the impulse imparted to the ball. Impulse is the change in momentum which equals the final momentum minus the initial momentum. Since the ball was initially at rest, its initial momentum is zero. To find the final momentum, we multiply the mass of the ball by the final velocity (its vector components). The impulse is then equal to the final momentum.

Find Impulse Imparted to the Club

Then we need to find the impulse imparted to the club. By the third law of motion, every action has an equal and opposite reaction, so the impulse imparted to the club is equal and opposite to the impulse imparted to the ball. Therefore, the magnitude of the impulse would be the same but the direction would be reversed.

Calculate Average Force Exerted on the Ball

Lastly, we calculate the average force exerted on the ball by the club. Force is equal to the change in momentum divided by the change in time, or basically, impulse over time. We divide the magnitude of impulse we found in step 2 by the time given to find the average force exerted on the ball by the club.

Key concepts

Understanding the Impulse-Momentum Theorem

In the realm of physics, the impulse-momentum theorem is a powerful tool for analyzing the interactions between objects. This theorem states that the impulse on an object is equal to the change in its momentum. Impulse can be visualized as a force applied over a period of time, and momentum is the product of an object's mass and velocity.

Applying this to a practical situation, like a golfer striking a ball, allows us to understand the effects of that impact. If the ball was at rest before being hit, its initial momentum is zero. The final momentum after the hit is calculated by multiplying the ball's mass by its final velocity. Since impulse equals the change in momentum, in this case, the impulse is simply the final momentum of the ball.

In the given exercise, after determining the final velocity components using trigonometry, the impulse imparted to the golf ball is found by using the formula: \[ \text{Impulse} = \text{Mass} \times \text{Final Velocity} \] The direction of this impulse is the same as the direction of the final velocity - in this case, upwards and forwards at a 30-degree angle.

Calculating the Average Force

Once we know the impulse exerted on an object, calculating the average force during that interaction is straightforward. The average force is simply the total impulse divided by the time interval over which it acts.

Using this concept in our golf ball scenario, we can determine the average force exerted by the club during the brief contact with the ball. To calculate the average force, we employ the formula: \[ \text{Average Force} = \frac{\text{Impulse}}{\text{Time Interval}} \] Since the time interval is given in milliseconds, we should convert it to seconds to maintain SI unit consistency. This gives us an accurate measure of the force exerted on the ball, which is a crucial element in evaluating the effectiveness of the golf swing.

Applying Trigonometry in Physics

Trigonometry often plays a vital role in solving physics problems, as it allows us to break down vectors into their components. When an object moves at an angle, as with a golf ball hit at a 30-degree incline, it has both horizontal and vertical components of velocity.

In the exercise, trigonometric functions help us resolve the initial velocity of the golf ball into its x-component (horizontal) and y-component (vertical) using cosine and sine functions, respectively:

• For the horizontal component: \[ v_{fx} = 52.2 \cdot \cos(30^\circ) \]
• For the vertical component: \[ v_{fy} = 52.2 \cdot \sin(30^\circ) \]

Understanding these components is essential for analyzing the motion of the ball post-impact, as well as for determining the influence of gravity and eventual trajectory of the ball.