/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 28 In a tennis match, a player wins... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 28

In a tennis match, a player wins a point by hitting the ball sharply to the ground on the opponent's side of the net. (a) If the ball bounces upward from the ground with a speed of \(16 \mathrm{m} / \mathrm{s}\) and is caught by a fan in the stands with a speed of \(12 \mathrm{m} / \mathrm{s}\), how high above the court is the fan? Ignore air resistance. (b) Explain why it is not necessary to know the mass of the tennis ball.

Short Answer

Expert verified
Height above court is approximately 5.71 meters. Mass is not needed because it cancels out in energy conservation.

Step by step solution

01

Understand the Problem

We need to find the height (h) where the speed of the tennis ball decreases from its initial speed of 16 m/s to a final speed of 12 m/s due to gravitational effect. We ignore air resistance.
02

Use Energy Conservation

The total mechanical energy (kinetic + potential energy) in a system remains constant if only conservative forces (like gravity) are doing work. Apply the principle:At the point of bounce:\[ KE_i = \frac{1}{2}m(16)^2, \ PE_i = 0 \]At the height where caught:\[ KE_f = \frac{1}{2}m(12)^2, \ PE_f = mgh \]Apply the conservation of energy equation:\[ \frac{1}{2}m(16)^2 = \frac{1}{2}m(12)^2 + mgh \]
03

Simplify and Solve for Height

Cancel mass (m) because it is present in every term of the equation:\[ \frac{1}{2}(16)^2 = \frac{1}{2}(12)^2 + gh \]Simplify further:\[ 128 = 72 + 9.8h \]Calculate h:\[ 9.8h = 56 \]\[ h = \frac{56}{9.8} \approx 5.71 \text{ meters} \]
04

Explain Importance of Mass

In part (b), the mass is not needed because it cancels out when using the energy conservation equation. The energy terms involve mass, but since it's in every term, it does not affect the final calculation of height.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Kinetic Energy
Kinetic energy is a type of energy that an object possesses due to its motion. It depends on the mass and velocity of the moving object. The formula for kinetic energy is given by:
  • Formula: \( KE = \frac{1}{2}mv^2 \)
  • Where \( m \) is the mass and \( v \) is the velocity of the object
In the tennis exercise, when the ball hits the ground and rebounds, it has kinetic energy because it is moving. Initially, as the ball moves from the ground with a speed of 16 m/s, its kinetic energy can be calculated using the given speed, although the mass doesn't matter for our final result. The decrease in speed from 16 m/s to 12 m/s as it is caught by the fan means that some kinetic energy has been transformed into potential energy as it rises.
Potential Energy
Potential energy is the energy stored in an object due to its position relative to a reference point, often in a gravitational field. The potential energy due to height is specifically called gravitational potential energy. The formula is given by:
  • Formula: \( PE = mgh \)
  • Where \( m \) is mass, \( g \) is acceleration due to gravity (approximately 9.8 m/s虏 on Earth), and \( h \) is the height above the reference point
In the tennis ball problem, as the ball rises after bouncing off the ground, it accumulates potential energy. When the ball is caught by the fan at some height, this potential energy will have increased compared to just after the bounce, even though the mass is impertinent in final calculations. As the ball's speed decreases from 16 m/s to 12 m/s, its kinetic energy decreases while its potential energy increases due to the gain in height.
Gravitational Effects
Gravity plays a crucial role in the motion of objects near the Earth鈥檚 surface. It provides the attractive force that pulls all objects toward the center of the Earth. Gravitational effects are not dependent on the object's mass when considering energy conservation for this tennis exercise.
  • Gravitational acceleration (\( g \)) is approximately 9.8 m/s虏.
  • All objects fall at the same rate, regardless of mass, when air resistance is negligible.
In the context of the exercise, gravity is the only force acting on the tennis ball during its trajectory from the bounce to when it is caught. This force is what transforms the ball's kinetic energy into potential energy. As the ball rises, it slows down because gravity opposes its upward motion, hence reducing kinetic energy and increasing potential energy based on height.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Jeff of the Jungle swings on a \(7.6-\mathrm{m}\) vine that initially makes an angle of \(37^{\circ}\) with the vertical. If Jeff starts at rest and has a mass of \(78 \mathrm{kg}\), what is the tension in the vine at the lowest point of the swing?

A person is to be released from rest on a swing pulled away from the vertical by an angle of \(20.0^{\circ} .\) The two frayed ropes of the swing are \(2.75 \mathrm{m}\) long, and will break if the tension in either of them exceeds \(355 \mathrm{N}\). (a) What is the maximum weight the person can have and not break the ropes? (b) If the person is released at an angle greater than \(20.0^{\circ}\), does the maximum weight increase, decrease, or stay the same? Explain.

A trapeze artist of mass \(m\) swings on a rope of length \(L\) Initially, the trapeze artist is at rest and the rope makes an angle \(\theta\) with the vertical. (a) Find the tension in the rope when it is vertical. (b) Explain why your result for part (a) depends on \(L\) in the way it does.

Two blocks, each of mass \(m\), are connected on a frictionless horizontal table by a spring of force constant \(k\) and equilibrium length \(L\). Find the maximum and minimum separation between the two blocks in terms of their maximum speed, \(v_{\max }\), relative to the table. (The two blocks always move in opposite directions as they oscillate back and forth about a fixed position.)

At the local playground a child on a swing has a speed of \(2.02 \mathrm{m} / \mathrm{s}\) when the swing is at its lowest point. (a) To what maximum vertical height does the child rise, assuming he sits still and "coasts"? Ignore air resistance. (b) How do your results change if the initial speed of the child is halved?
See all solutions

Recommended explanations on Physics Textbooks

Further Mechanics and Thermal Physics

Read Explanation

Modern Physics

Read Explanation

Kinematics Physics

Read Explanation

Magnetism

Read Explanation

Measurements

Read Explanation

Circular Motion and Gravitation

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.