Problem 39 - College Physics

Question

. Biomechanics. The mass of a regulation tennis ball is 57 g (although it can vary slightly), and tests have shown that the ball is in contact with the tennis racket for 30 ms. (This number can also vary, depending on the racket and swing.) We shall assume a 30.0 \(\mathrm{ms}\) contact time throughout this problem. The fastest-known served tennis ball was served by "Big Bill" Tilden in \(1931,\) and its speed was measured to be 73.14 \(\mathrm{m} / \mathrm{s}\) .(a) What impulse and what force did Big Bill exert on the ten- nis ball in his record serve? (b) If Big Bill's opponent returned his serve with a speed of \(55 \mathrm{m} / \mathrm{s},\) what force and what impulse did he exert on the ball, assuming only horizontal motion?

Step-by-Step Solution

Answer

(a) Impulse: 4.169 kg m/s, Force: 139 N. (b) Impulse: 7.304 kg m/s, Force: 243 N.

Step 1: Understanding Impulse

Impulse (J) is the change in momentum of an object. It can be calculated using the formula: \[J = \Delta p = m\Delta v\]where \(m\) is the mass and \(\Delta v \) is the change in velocity of the object.

Step 2: Calculating Bill's Impulse

Given the initial velocity (v_i) of the tennis ball is 0 (since it is stationary before the serve), and the final velocity (v_f) is 73.14 \mathrm{m/s}, the change in velocity (\Delta v) is: \[73.14 \mathrm{m/s} - 0 = 73.14 \mathrm{m/s}\]The mass (m) of the ball is 57 g, or 0.057 kg. Using the impulse formula: \[J = m\Delta v = 0.057 \times 73.14 = 4.16898 \text{ kg m/s}\]

Step 3: Calculating Force from Bill's Serve

Force (F) can be calculated using the impulse-momentum theorem: \[F \cdot \Delta t = J\]where \(\Delta t\) is the contact time (30 ms = 0.030 s). Solving for \(F\), we have: \[F = \frac{J}{\Delta t} = \frac{4.16898}{0.030} = 138.966 \text{ N}\]

Step 4: Calculating Opponent's Impulse

The opponent returns the ball with a speed of 55 \mathrm{m/s}. Assuming the direction is opposite to Big Bill's serve, the change in velocity is: \[(v_f = 55 \text{ m/s, opposite direction})\Delta v = 0 \text{ m/s} - (-73.14 \text{ m/s}) = 73.14 + 55 = 128.14 \text{ m/s}\]Then the impulse is: \[J = m\Delta v = 0.057 \times 128.14 = 7.30398 \text{ kg m/s}\]

Step 5: Calculating Force from Opponent's Return

Using the same method as Step 3 and the same contact time, the force exerted by the opponent is: \[F = \frac{J}{\Delta t} = \frac{7.30398}{0.030} = 243.466 \text{ N}\]

Key Concepts

Understanding Impulse

Impulse is a key concept in biomechanics, especially when analyzing the motion of objects like a tennis ball during a serve. When we talk about impulse, we mean the change in momentum that an object experiences when a force is applied over a period of time. Think of it as a way to measure how much a push or hit changes the motion of the ball.

Impulse can be calculated using the formula:

• \[ J = \Delta p = m \Delta v \]
• Here, \( m \) denotes the mass of the object, and \( \Delta v \) represents the change in velocity.

So, to find the impulse experienced by a tennis ball during a rapid serve, you simply multiply the ball's mass by how much its velocity changes during the contact with the racket. This helps us understand how powerful Big Bill Tilden's serve was, by quantifying the effect it had on the ball's motion.

Grasping Momentum

Momentum is another crucial part of understanding biomechanics, especially in sports like tennis. It's the quantity that describes an object's motion, depending on both its mass and how fast it's moving. Simply put, momentum is the product of mass and velocity. If an object has a lot of momentum, it means it's moving quickly or is very heavy, or both!

In formulaic terms:

• \[ p = m \times v \]
• where \( p \) is momentum, \( m \) is mass, and \( v \) is velocity.

In the case of Big Bill's serve, his tennis ball needed to have a significant change in momentum to reach that staggering speed. This change is captured neatly by impulse—as seen via the impulse-momentum theorem. It tells us that the resulting change in momentum equals the impulse applied, which blends the effects of force and time.

The Role of Force

Force is a push or pull that acts upon an object due to its interaction with another object. In tennis, when you strike the ball with a racket, you're applying force. The impulse-momentum theorem connects impulse and force by showing how applying force over time changes an object's momentum.

In the exercise, Big Bill applied a powerful force on the tennis ball during his serve. We can calculate this force using:

• \[ F \cdot \Delta t = J \]
• where \( F \) is the force and \( \Delta t \) is the contact time.

Solving for force gives us \[ F = \frac{J}{\Delta t} \]. Essentially, if we know the impulse and the time it took to apply that force, we can find out exactly how hard Big Bill hit the ball. Forces cause changes in motion, making them central to understanding how things move in sports.

Understanding Contact Time

Contact time is the duration for which two objects are in contact during an interaction. In biomechanics, particularly in sports like tennis, contact time is vital in understanding how force and impulse work together to alter an object's momentum. The longer the contact, the more it softens the transfer of force, while still achieving the same change in momentum.

For Big Bill's record-breaking tennis serve, the ball was in contact with the racket for a brief 30 milliseconds. But even in such a short time frame, a significant change in speed was achieved thanks to the force applied. The impulse-momentum theorem uses contact time to show that force affects how long and how effectively a mass gets its velocity changed.

• Think of it like a follow-through in a golf swing or a baseball bat hitting a ball: the longer the club or bat stays with the ball, the greater the effect of the force.

Understanding contact time can help athletes adjust their techniques to maximize their performance.