Question

Tennis champion Venus Williams is capable of serving a tennis ball at around 127 mph. a) Assuming that her racquet is in contact with the \(57.0-\mathrm{g}\) ball for \(0.250 \mathrm{~s}\), what is the average force of the racquet on the ball? b) What average force would an opponent's racquet have to exert in order to return Williams's serve at a speed of \(50.0 \mathrm{mph}\), assuming that the opponent's racquet is also in contact with the ball for \(0.250 \mathrm{~s} ?\)

Short answer

Answer: The average force exerted by Venus Williams's racquet on the ball when serving is 12.95 N, and the average force exerted by an opponent's racquet to return the serve at a speed of 50 mph is 18.05 N.

Step-by-step solution

(Step 1: Convert given velocities to m/s)

Convert the given velocity values from mph to m/s. We will use the conversion factor: 1 mph = 0.44704 m/s (a) Venus Williams's serve: Initial velocity (\(u\)) = 0 (at rest) Final velocity (\(v\)) = 127 mph * 0.44704 m/s = 56.8 m/s (b) Opponent's return serve: Initial velocity (\(u\)) = -56.8 m/s (opposite direction) Final velocity (\(v\)) = 50 mph * 0.44704 m/s = 22.35 m/s

(Step 2: Calculate acceleration for each case)

For each case, use the formula \(a = \frac{v - u}{t}\) to find the acceleration. (a) \(a = \frac{56.8 - 0}{0.25} = 227.2 \, \text{m/s}^2\) (b) \(a = \frac{22.35 - (-56.8)}{0.25} = 316.6 \, \text{m/s}^2\)

(Step 3: Calculate average force)

Using the calculated acceleration and Newton's second law, we can find the average force exerted. (a) \(F = m \cdot a = 0.057 \, \text{kg} \cdot 227.2 \, \text{m/s}^2 = 12.95 \, \text{N}\) (b) \(F = m \cdot a = 0.057 \, \text{kg} \cdot 316.6 \, \text{m/s}^2 = 18.05 \, \text{N}\)

(Step 4: Final Answers)

(a) The average force exerted by Venus Williams's racquet on the ball is \(12.95 \, \text{N}\). (b) The average force an opponent's racquet would have to exert in order to return Williams's serve at a speed of \(50.0 \mathrm{mph}\) is \(18.05 \, \text{N}\).

Key concepts

Newton's Second Law

Newton’s Second Law of Motion is a fundamental principle that describes the relationship between an object's mass, its acceleration, and the applied force. This law can be expressed with the simple formula:
\[ F = m \times a \]
where \( F \) is the force applied to an object, \( m \) is the mass of the object, and \( a \) is the object's acceleration. The units of force are Newtons (N), which can be broken down into kilograms times meters per second squared (\( kg \times m/s^2 \)).

In the context of a tennis serve, Newton's Second Law allows us to calculate the average force exerted by Venus Williams's racquet on the tennis ball. By knowing the mass of the ball and measuring its acceleration as it leaves the racquet, we can determine the force that was applied during the serve. This illustration of Newton's Second Law shows that the greater the acceleration (or the mass), the greater the force needed - which makes sense when considering a powerful tennis serve.

Acceleration Calculation

Acceleration is defined as the rate of change of velocity of an object. Using the following equation, we can calculate the acceleration (\( a \)):
\[ a = \frac{v - u}{t} \]
where \( v \) is the final velocity, \( u \) is the initial velocity, and \( t \) is the time over which this change occurs. The units of acceleration are meters per second squared (\( m/s^2 \)).

For Venus Williams's serve, we saw her racquet bring the ball from rest to a high velocity within a fraction of a second, resulting in a significant acceleration. Similarly, an opponent would also apply a force over a short time to return the serve, creating another instance where acceleration is critical. Calculating the exact values for each situation enables us to apply Newton's Second Law and determine the average forces involved in each serve and return.

Velocity Conversion

In our global world, it’s common to encounter different units of measurement. When studying motion, we often need to convert velocities between units to ensure consistency within calculations. Meters per second (\( m/s \)) is the standard unit of velocity in the International System of Units (SI), which is why we convert the velocity from miles per hour (mph) using the conversion factor:
\[ 1 \, mph = 0.44704 \, m/s \]
When working with Venus Williams's serve velocity, the conversion from 127 mph to meters per second was necessary to calculate acceleration correctly. The same conversion was needed for the opponent's return velocity. This step ensures that all velocities are expressed in the same units, allowing for accurate and comparable force calculations.