Problem 26 - Thomas Calculus

Question

Tennis \(A\) 2-oz tennis ball was served at 160 \(\mathrm{ft} /\) sec (about 109 \(\mathrm{mph}\) ). How much work was done on the ball to make it go this fast? (To find the ball's mass from its weight, express the weight in pounds and divide by \(32 \mathrm{ft} / \mathrm{sec}^{2},\) the acceleration of gravity.)

Step-by-Step Solution

Answer

The work done is approximately 50 ft-lbs.

Step 1: Convert the Weight to Pounds

The weight of the tennis ball is given as 2 ounces. We will first convert this to pounds. Since 1 pound equals 16 ounces, we have: \[ \text{Weight in pounds} = \frac{2 \text{ oz}}{16} = 0.125 \text{ lbs} \]

Step 2: Calculate the Mass of the Ball

Use the weight in pounds to calculate the mass. Mass is calculated by dividing the weight by the acceleration due to gravity (32 ft/s²): \[ \text{Mass} = \frac{0.125 \text{ lbs}}{32 \text{ ft/s}^2} \approx 0.00390625 \text{ slugs} \]

Step 3: Use Kinetic Energy Formula

The work done on the tennis ball to bring it to the serving speed of 160 ft/sec can be equated to its kinetic energy at that speed. Kinetic energy is given by the formula: \[ KE = \frac{1}{2}mv^2 \]where \( m \) is mass and \( v \) is velocity. Substituting the values: \[ KE = \frac{1}{2} \times 0.00390625 \text{ slugs} \times (160 \text{ ft/s})^2 \]

Step 4: Compute Kinetic Energy (Work Done)

Now calculate the work done:\[ KE = \frac{1}{2} \times 0.00390625 \times 25600 \approx 50 \text{ ft-lbs} \]This value represents the amount of work done on the tennis ball to accelerate it to the given speed.

Key Concepts

Kinetic Energy

Kinetic energy is a fundamental concept in physics that describes the energy an object possesses due to its motion. When an object moves, it can do work on other objects and this is quantified by its kinetic energy.
Kinetic energy is calculated using the formula:

• \[ KE = \frac{1}{2}mv^2 \]
• Here, \( KE \) represents kinetic energy, \( m \) is the mass of the object, and \( v \) is the velocity.

In the context of our original exercise, the tennis ball was served to move at a speed of 160 ft/sec. By substituting the mass and velocity into the kinetic energy equation, we can calculate the work done to accelerate the ball to this speed. This is because work done in accelerating the ball translates directly into its kinetic energy at the given velocity.

Unit Conversion

Unit conversion is an important aspect when solving physics problems involving different measurement units.
In our exercise, the process starts with the conversion of the tennis ball's weight from ounces to pounds. This is necessary because standard equations in physics often require weights to be in consistent units:

• Weight in pounds is calculated as \( \frac{2 \text{ oz}}{16} = 0.125 \text{ lbs} \).

After obtaining the weight in pounds, it is crucial for subsequent calculations involving force or mass. Unit conversion ensures all the quantities are aligned with standard physics units, facilitating the computation of kinetic energy and work.

Mass and Weight

Understanding the difference between mass and weight is key in physics.

• Weight is the force exerted by gravity on an object, while mass is the measure of the amount of matter in that object.
• Weight depends on the gravitational pull and is calculated as the product of mass and gravitational acceleration.

In computing the mass from the weight of the tennis ball (given in pounds), we divide the weight by the acceleration due to gravity. This gives us the mass in slugs:

• \( \text{Mass} = \frac{0.125 \text{ lbs}}{32 \text{ ft/s}^2} \).

The distinction between mass and weight is fundamental, particularly in calculations that involve the acceleration due to gravity.

Acceleration Due to Gravity

The acceleration due to gravity is a constant that appears in many physics equations.

• On Earth, its value is approximately \( 32 \text{ ft/s}^2 \) or \( 9.8 \text{ m/s}^2 \) depending on the unit system.
• This constant converts weight (a force) to mass for calculations, ensuring accurate results in physics problems.

In our exercise, understanding gravity’s role allows the conversion of the tennis ball's weight into mass, which then plays a crucial part in determining how much work is done on the ball to reach its serving speed. Gravitational acceleration thus bridges the transition from gravitational force (weight) to inertial properties (mass).