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Chapter 4: Problem 38

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.

Short Answer

Expert verified
(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 by step solution

01

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: \[ t = \frac{30.0 \text{ ms}}{1000} = 0.030 \text{ s} \]
02

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 \( a \): \[ a = \frac{73.14}{0.030} = 2438 \text{ m/s}^2 \]
03

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.
04

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.
05

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

A small car of mass 380 kg is pushing a large truck of mass 900 kg due east on a level road. The car exerts a horizontal force of 1600 N on the truck. What is the magnitude of the force that the truck exerts on the car?

A box rests on a frozen pond, which serves as a frictionless horizontal surface. If a fisherman applies a horizontal force with magnitude 48.0 N to the box and produces an acceleration of magnitude 2.20 m/s\(^2\), what is the mass of the box?

An 8.00-kg box sits on a level floor. You give the box a sharp push and find that it travels 8.22 m in 2.8 s before coming to rest again. (a) You measure that with a different push the box traveled 4.20 m in 2.0 s. Do you think the box has a constant acceleration as it slows down? Explain your reasoning. (b) You add books to the box to increase its mass. Repeating the experiment, you give the box a push and measure how long it takes the box to come to rest and how far the box travels. The results, including the initial experiment with no added mass, are given in the table: In each case, did your push give the box the same initial speed? What is the ratio between the greatest initial speed and the smallest initial speed for these four cases? (c) Is the average horizontal force \(f\) exerted on the box by the floor the same in each case? Graph the magnitude of force \(f\) versus the total mass \(m\) of the box plus its contents, and use your graph to determine an equation for \(f\) as a function of \(m\).

A ball is hanging from a long string that is tied to the ceiling of a train car traveling eastward on horizontal tracks. An observer inside the train car sees the ball hang motionless. Draw a clearly labeled free-body diagram for the ball if (a) the train has a uniform velocity and (b) the train is speeding up uniformly. Is the net force on the ball zero in either case? Explain

A 68.5-kg skater moving initially at 2.40 m/s on rough horizontal ice comes to rest uniformly in 3.52 s due to friction from the ice. What force does friction exert on the skater?
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