Problem 38 - University Physics with Modern Physics

Question

The fastest served tennis ball, served by "Big Bill" Tilden in 1931, was measured at 73.14 m/s. The mass of a tennis ball is 57 g, and the ball, which starts from rest, is typically in contact with the tennis racquet for 30.0 ms. Assuming constant acceleration, (a) what force did Big Bill's tennis racquet exert on the ball if he hit it essentially horizontally? (b) Draw free-body diagrams of the ball during the serve and just after it moved free of the racquet.

Step-by-Step Solution

Answer

(a) The force exerted by the racquet is approximately 139 N. (b) During contact, the free-body diagram shows a larger force to the right and gravity downward; after contact, only gravity acts.

Step 1: Convert Units

First, we need to convert the mass of the tennis ball from grams to kilograms. Since there are 1000 grams in a kilogram, the mass is: \[ m = \frac{57\text{ g}}{1000} = 0.057 \text{ kg} \] Convert the time from milliseconds to seconds: \[ t = \frac{30.0 \text{ ms}}{1000} = 0.030 \text{ s} \]

Step 2: Calculate Acceleration

Using the formula for constant acceleration, \( v = u + at \), where \( v \) is the final velocity, \( u \) is the initial velocity (0 m/s), \( a \) is the acceleration, and \( t \) is the time.Rearrange the formula to solve for \( a \): \[ a = \frac{v - u}{t} = \frac{73.14 \text{ m/s} - 0 \text{ m/s}}{0.030 \text{ s}} \]Calculate \( a \): \[ a = \frac{73.14}{0.030} = 2438 \text{ m/s}^2 \]

Step 3: Calculate the Force

Using Newton's second law, \( F = ma \), where \( F \) is the force, \( m \) is the mass, and \( a \) is the acceleration. Substituting the known values:\[ F = 0.057 \text{ kg} \times 2438 \text{ m/s}^2 \]Calculate \( F \): \[ F = 138.966 \text{ N} \]Thus, the force exerted by the racquet on the ball is approximately 139 N.

Step 4: Free-Body Diagram During Contact

During the contact period, draw a free-body diagram of the tennis ball. The force diagram should show:- A large arrow pointing to the right labeled \( F \) (force from the racquet)- A small arrow pointing down labeled \( mg \) (weight of the ball due to gravity). The force from the racquet is much larger than the gravitational force during contact.

Step 5: Free-Body Diagram After Contact

After the ball leaves the racquet, the only force acting on the ball is gravity. Draw a diagram with:- A single downwards arrow labeled \( mg \) representing the gravitational force.- No horizontal forces acting on the ball once it is in free motion.

Key Concepts

Constant Acceleration

Constant acceleration occurs when an object's velocity changes at a consistent rate over time. In physics exercises, particularly those involving Newton's Second Law, this concept is essential for studying motion.

In our exercise, the tennis ball's velocity increases from rest (0 m/s) to 73.14 m/s in 0.030 seconds. This change in velocity happens at a steady rate, meaning the acceleration remains consistent throughout the contact with the racquet. The formula for constant acceleration, \( a = \frac{v - u}{t} \), is used to calculate the rate of acceleration.

By applying this formula, you can easily solve for the acceleration using the given velocities and time, leading you to understand how force affects motion under constant acceleration.

Force Calculation

The calculation of force is pivotal in understanding how objects interact with one another, especially under Newton's Second Law. This law states that the force exerted on an object is equal to the mass of the object multiplied by its acceleration, expressed mathematically as \( F = ma \).

In the exercise, we use the derived constant acceleration \( 2438 \text{ m/s}^2 \) and the mass of the tennis ball, \( 0.057 \text{ kg} \), to find the force exerted by the racquet. Substituting these values into the equation illustrates how a relatively small mass can experience a significant force when subjected to high acceleration.

Understanding force calculation helps in recognizing how varying forces influence an object's motion, making it one of the cornerstones of classical mechanics.

Free-Body Diagrams

Free-body diagrams are visual tools used to represent the forces acting upon an object. They help simplify complex force interactions into understandable visuals.

When the tennis ball is in contact with the racquet, the free-body diagram shows two forces. A larger horizontal force \( F \) to the right from the racquet and a smaller vertical force \( mg \), the ball's weight, pointing down. This illustrates the dominance of the horizontal force over gravity during contact.

Once free from the racquet, the only acting force is gravity, shown as a single downward arrow \( mg \). This change in the diagram reflects the transition from applied force control to free fall, highlighting gravity as the sole influence once the ball leaves the racquet. Free-body diagrams thus serve as a crucial means to depict and analyze forces clearly and effectively.