Problem 36 - College Physics

Question

Biomechanics. The mass of a regulation tennis ball is \(57.0 \mathrm{~g}\) (although it can vary slightly), and tests have shown that the ball is in contact with the tennis racket for \(30 \mathrm{~ms}\). (This number can also vary, depending on the racket and swing.) We assume a \(30.0 \mathrm{~ms}\) contact time in this problem. In the 2011 Davis Cup competition, Ivo Karlovic made one of the fastest recorded serves in history, which was clocked at \(156 \mathrm{mph}(70 \mathrm{~m} / \mathrm{s}) .\) (a) What impulse and what average force did Karlovic exert on the tennis ball in his record serve? (b) If his opponent returned this serve with a speed of \(55.0 \mathrm{~m} / \mathrm{s},\) what impulse and what average force did his opponent exert on the ball, assuming purely horizontal motion?

Step-by-Step Solution

Answer

(a) Impulse: 3.99 kg·m/s, Force: 133 N. (b) Impulse: 7.125 kg·m/s, Force: 238 N.

Step 1: Convert Units of Mass

The mass of the tennis ball is given in grams. Convert this to kilograms because SI units should be used. \(57.0\, \text{g} = 0.057\, \text{kg}\).

Step 2: Determine Change in Velocity

For Karlovic's serve, the final velocity \(v_f\) is \(70\, \text{m/s}\) and the initial velocity \(v_i\) is \(0\, \text{m/s}\) since the ball was at rest before being hit.

Step 3: Calculate Impulse (a)

Impulse is the change in momentum. Use the formula \(\Delta p = m \cdot (v_f - v_i)\). Substituting the given values: \[\Delta p = 0.057\, \text{kg} \cdot (70\, \text{m/s} - 0\, \text{m/s}) = 3.99\, \text{kg} \cdot \text{m/s}\].

Step 4: Determine Average Force (a)

Average force is calculated using the formula \(F_{avg} = \frac{\Delta p}{\Delta t}\). Substituting \(\Delta p = 3.99\, \text{kg} \cdot \text{m/s}\) and \(\Delta t = 0.030\, \text{s}\): \[F_{avg} = \frac{3.99}{0.030} = 133\, \text{N}\].

Step 5: Calculate Impulse (b)

For the opponent's return, the final velocity \(v_f\) is \(55\, \text{m/s}\) and the initial velocity \(v_i\) is the negative of Karlovic's serve because it's in the opposite direction, thus \(-70\, \text{m/s}\). Using the impulse formula, \[\Delta p = 0.057\, \text{kg} \cdot (55\, \text{m/s} - (-70\, \text{m/s})) = 7.125\, \text{kg} \cdot \text{m/s}\].

Step 6: Determine Average Force (b)

Substitute \(\Delta p = 7.125\, \text{kg} \cdot \text{m/s}\) and \(\Delta t = 0.030\, \text{s}\) into the average force formula: \[F_{avg} = \frac{7.125}{0.030} = 238\, \text{N}\].

Key Concepts

Impulse in Physics

Impulse is a fundamental concept in physics, closely tied to the change in an object's momentum. Imagine a tennis ball being struck by a racket; the force exerted over the time the ball is in contact with the racket is what we refer to as impulse.
In simple terms, impulse can be defined as the product of the average force applied on an object and the time duration over which this force is applied. The formula for impulse \( J \) is given by:

• \( J = F_{avg} \times \Delta t \)

where:

• \( F_{avg} \) is the average force
• \( \Delta t \) is the time duration

Impulse also equals the change in momentum \( \Delta p \) of the object. Mathematically, it's expressed as:

• \( J = \Delta p = m \cdot (v_f - v_i) \)

where:

• \( m \) is the mass of the object
• \( v_f \) and \( v_i \) are the final and initial velocities

This concept is crucial in sports, such as tennis, where the speed and direction of a ball change significantly due to the racket's force.

Average Force Calculation

Calculating the average force involves understanding how force contributes to changes in momentum during the time of collision with an object. In our tennis example, during Karlovic's powerful serve, the racket exerts a continuous force on the ball while they are in contact for a very brief period. To find this average force \( F_{avg} \), you need the change in momentum \( \Delta p \) and the time duration \( \Delta t \):

• \( F_{avg} = \frac{\Delta p}{\Delta t} \)

The change in momentum for Karlovic’s serve is calculated as \( 3.99 \, \text{kg} \cdot \text{m/s} \). The force over the incredibly short interaction time of \( 0.030 \, \text{s} \) yields:

• \( F_{avg} = \frac{3.99}{0.030} = 133 \, \text{N} \)

This equation reveals the significant force exerted briefly to achieve the tennis ball's high-speed motion. Understanding this helps in appreciating the physical demands and precision involved in sports such as tennis.

Momentum in Sports Physics

Momentum, a vector quantity in physics, is simply the product of an object's mass and velocity. In sports physics, tracking and altering momentum is key to understanding and improving performance. Let's highlight how momentum plays a role in tennis, a sport where managing speeds and directions is crucial.
Momentum \( p \) is given by:

• \( p = m \times v \)

where:

• \( m \) is the mass, and \( v \) is the velocity of the object

In our scenario, momentum involves the change when the ball is served and subsequently returned. Karlovic's serve gives the ball a substantial initial momentum by accelerating it from rest to a high speed. When the opponent returns the serve, the direction of momentum is reversed, and its magnitude is considerably affected. The return involves a momentum change from \( 0.057 \, \text{kg} \times -70 \, \text{m/s} \) to \( 0.057 \, \text{kg} \times 55 \, \text{m/s} \), resulting in a significant change.
Understanding momentum changes enables athletes to exert the right amount of force and timing to efficiently control the ball's speed and direction during plays.