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Human biomechanics. The fastest served tennis ball, served by Samuel Groth in \(2012,\) was measured at \(73 \mathrm{~m} / \mathrm{s}\). The mass of a tennis ball is \(57 \mathrm{~g}\), and the ball is typically in contact with the tennis racquet for \(30.0 \mathrm{~ms}\), with the ball starting from rest. Assuming constant acceleration, (a) what force did Groth's tennis racquet exert on the tennis ball if he hit it essentially horizontally? (b) Make a free-body diagram of the tennis ball during the serve and one just after it has moved free of the racquet.

Short Answer

Expert verified
The force exerted by the racquet is 138.7 N. During the serve, the tennis ball experiences the force of the racquet and its weight. After the serve, only the weight acts on it.

Step by step solution

01

Convert Units to Standard SI Units

The mass of the tennis ball is given as \( 57 \) g, which needs to be converted to kilograms. Since \( 1 \) kg \( = 1000 \) g, the mass of the tennis ball in kilograms is \( 0.057 \) kg. The time of contact is given as \( 30.0 \) ms, which in seconds is \( 0.030 \) s. The speed is already given in \( \text{m/s} \), which is an appropriate unit for this calculation.
02

Calculate Acceleration

Use the formula for constant acceleration, \( a = \frac{v - u}{t} \), where \( v = 73 \) m/s is the final velocity, \( u = 0 \) m/s is the initial velocity, and \( t = 0.030 \) s is the time.Substitute the given values: \[a = \frac{73 - 0}{0.030} = \frac{73}{0.030} = 2433.33 \text{m/s}^2.\]
03

Calculate the Force Exerted

Use Newton's second law of motion, \( F = ma \), where \( m = 0.057 \) kg is the mass, and \( a = 2433.33 \text{m/s}^2 \) is the acceleration.Substitute the values to find the force: \[F = 0.057 \times 2433.33 = 138.7 \text{N}.\]
04

Draw Free-Body Diagrams

1. **During the Serve:** The forces acting on the tennis ball are the force exerted by the racquet (\( F \)), directed horizontally, and the weight of the ball (\( mg \)), directed vertically downward.2. **After the Serve:** Once the ball is free from the racquet, the only force acting on it is the gravitational force (weight \( mg \)), directed downward.

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Key Concepts

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

Newton's Second Law
Newton's Second Law of Motion is a cornerstone concept in physics that describes how an object’s motion changes when it is subjected to external forces. The law can be expressed mathematically as \( F = ma \), where \( F \) represents force, \( m \) stands for mass, and \( a \) signifies acceleration.
In the context of human biomechanics, particularly in sports activities like a tennis serve, this law helps us understand how the force applied by a tennis racquet to a ball results in the ball's acceleration.
The force and acceleration are directly related, meaning that the more force applied (given a constant mass), the greater the acceleration the object experiences.
Newton’s Second Law not only enables us to calculate the amount of force exerted on an object but also helps in predicting the resulting motion which is crucial in biomechanics calculations.
Force Calculation
To calculate the force exerted on an object, especially in a biomechanics scenario, we need the object’s mass and its acceleration. Using the formula \( F = ma \), you ensure the force calculation is precise and reflective of real-world scenarios.
For the tennis ball example, knowing the racket interacts with the ball over a small interval helps. The mass and derived acceleration from the constant velocity change are plugged into the formula.
This calculation gives insight into how much force athletes exert in their actions. Useful for trainers and athletes alike, who wish to optimize performance or understand physical limits, thus honing skills through data-backed knowledge.
Free-Body Diagram
Free-body diagrams are essential tools in physics and biomechanics for illustrating the forces acting on an object. They help visualize and simplify the problem, making it easier to solve complex force-related tasks.
When the tennis ball is in contact with the racket, the diagram would include two main forces: the horizontal force applied by the racket and the vertical gravitational force (the ball's weight).
Once the ball leaves the racket, only the gravitational force remains, pointing downward. Such diagrams are pivotal in understanding and solving dynamics problems, which is why they are often used in biomechanics assessments, especially in sports science.
Constant Acceleration
Constant acceleration occurs when an object’s velocity changes at a consistent rate over time. In the context of Newton's laws, and particularly for the tennis ball scenario, this assumption simplifies calculations.
For example, a consistent acceleration can result from a steady application of force, as seen when a racket hits a ball. Calculating this acceleration involves using the formula \( a = \frac{v - u}{t} \), where \( v \) is the final velocity, \( u \) is the initial velocity, and \( t \) is the time during which the force is applied.
The concept of constant acceleration allows for more straightforward mathematical models and predictions in biomechanics, ensuring easy replication and comparison across various studies or physical realities.
Unit Conversion
Unit conversion is a critical step in solving physics and biomechanics problems, ensuring calculations are performed in standardized units. Often, measurements may be given in a variety of units—such as grams for mass or milliseconds for time—which must be converted to the International System of Units (SI units) like kilograms and seconds.
In the tennis serve example, converting the tennis ball's mass from grams to kilograms (1 kg = 1000 g) and the contact time from milliseconds to seconds (1 s = 1000 ms) is necessary for accurate force and acceleration computations.
Unit conversion ensures consistency and accuracy, allowing for valid comparisons with theoretical models and experimental data. Mastering this skill is indispensable for students and professionals working in scientific fields.

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Most popular questions from this chapter

A tennis ball traveling horizontally at \(22 \mathrm{~m} / \mathrm{s}\) suddenly hits a vertica brick wall and bounces back with a horizontal velocity of \(18 \mathrm{~m} / \mathrm{s}\). Make a free-body diagram of this ball (a) just before it hits the wall, (b) just after it has bounced free of the wall, and (c) while it is in contact with the wall.

You drag a heavy box along a rough horizontal floor by a horizontal rope. Identify the reaction force to each of the following forces: (a) the pull of the rope on the box, (b) the friction force on the box, (c) the normal force on the box, and (d) the weight of the box.

A man is dragging a trunk up the loading ramp of a mover's truck. (See Figure \(4.37 .)\) The ramp has a slope angle of \(20.0^{\circ},\) and the man pulls upward with a force \(\vec{F}\) of magnitude \(375 \mathrm{~N}\) whose direction makes an angle of \(30.0^{\circ}\) with the ramp. Find the horizontal and vertical components of the force \(\vec{F}\)

Forces on a dancer's body. Dancers experience large forces associated with the jumps they make. For example, when a dancer lands after a vertical jump, the force exerted on the head by the neck must exceed the head's weight by enough to cause the head to slow down and come to rest. The head is about \(9.4 \%\) of a typical person's mass. Video analysis of a \(65 \mathrm{~kg}\) dancer landing after a vertical jump shows that her head slows down from \(4.0 \mathrm{~m} / \mathrm{s}\) to rest in a time of \(0.20 \mathrm{~s}\) What is the magnitude of the average force that her neck exerts on her head during the landing? A. \(0 \mathrm{~N}\) B. \(60 \mathrm{~N}\) C. \(120 \mathrm{~N}\) D. \(180 \mathrm{~N}\)

Two crates, \(A\) and \(B\), sit at rest side by side on a frictionless horizontal surface. The crates have masses \(m_{\mathrm{A}}\) and \(m_{\mathrm{B}}\). A horizontal force \(\vec{F}\) is applied to crate \(A,\) and the two crates move off to the right. (a) Draw clearly labeled free-body diagrams for crate \(A\) and for crate \(B\). Indicate which pairs of forces, if any, are third-law action-reaction pairs. (b) If the magnitude of force \(\overrightarrow{\boldsymbol{F}}\) is less than the total weight of the two crates, will it cause the crates to move? Explain.

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