/*! 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 15 A quarterback throws a pass that... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 15

A quarterback throws a pass that is a perfect spiral. In other words, the football does not wobble, but spins smoothly about an axis passing through each end of the ball. Suppose the ball spins at \(7.7 \mathrm{rev} / \mathrm{s} .\) In addition, the ball is thrown with a linear speed of \(19 \mathrm{~m} / \mathrm{s}\) at an angle of \(55^{\circ}\) with respect to the ground. If the ball is caught at the same height at which it left the quarterback's hand, how many revolutions has the ball made while in the air?

Short Answer

Expert verified
The ball makes approximately 24.41 revolutions in the air.

Step by step solution

01

Determine Time in Air

To find out how long the ball is in the air, we need to consider the vertical component of the throw and use the equation for projectile motion. The vertical velocity component is \( v_{y} = 19 \sin(55^{\circ}) \). The time of flight when caught at the same level is given by \( t = \frac{2v_y}{g} \), where \( g = 9.81 \, \text{m/s}^2 \). Calculate \( v_{y} \) and then \( t \).
02

Vertical Velocity Calculation

Calculate the vertical component of the initial velocity. Use \( v_y = 19 \sin(55^{\circ}) \). Solving gives \( v_y \approx 15.57 \, \text{m/s} \).
03

Compute Time of Flight

Using the formula \( t = \frac{2v_y}{g} \), substitute \( v_y = 15.57 \, \text{m/s} \) and \( g = 9.81 \, \text{m/s}^2 \) to find \( t \). This results in \( t \approx 3.17 \, \text{seconds} \).
04

Calculate Number of Revolutions

During the time the ball is in the air, it spins at a rate of \( 7.7 \text{ rev/s} \). To find the total number of revolutions, multiply the spin rate by the time in air: \( \text{Number of revolutions} = 7.7 \, \text{rev/s} \times 3.17 \, \text{s} \approx 24.41 \).

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.

Rotational Motion
Rotational motion is when an object rotates around a fixed axis. It's different from linear motion, where an object moves along a path without spinning. In this exercise, the football spins smoothly on its axis, making a perfect spiral. This kind of motion is typical in sports and helps maintain a stable flight path.

For analyzing rotational motion, we need to consider key concepts such as:
  • The axis of rotation, which is the imaginary line that the object spins around. In our case, it passes through the ends of the football.
  • Rotational speed, which is how fast the object spins. Here, the football spins at 7.7 revolutions per second.
  • The uniformity of the spin, which affects the motion's stability. A perfect spiral means the spin is consistent without wobbling.
These factors combined ensure the football travels accurately to its target.
Vertical Velocity
Vertical velocity refers to the speed of an object along the vertical axis. It plays a crucial role in determining how long an object stays in the air. In the context of projectile motion, it determines how quickly an object reaches its peak height and returns.

For the football pass:
  • Vertical velocity is calculated using the component of the initial speed in the vertical direction. This is done through the equation: \( v_{y} = v_0 \sin(\theta) \), where \( v_0 \) is the initial speed and \( \theta \) is the angle of the throw.
  • Given \( v_0 = 19 \, \text{m/s} \) and \( \theta = 55^{\circ} \), the vertical velocity \( v_{y} \) is approximately \( 15.57 \, \text{m/s} \).
This velocity tells us how aggressively the football rises, helping predict its time aloft.
Time of Flight
Time of flight is the duration an object remains in the air during projectile motion. This time is influenced by the object's vertical velocity and the force of gravity.

Calculating time of flight involves:
  • Using the formula \( t = \frac{2v_y}{g} \), where \( g \) is the acceleration due to gravity, approximately \( 9.81 \, \text{m/s}^2 \).
  • Substituting in our vertical velocity \( v_{y} = 15.57 \, \text{m/s} \), resulting in the time of flight being \( t \approx 3.17 \, \text{seconds} \).
Understanding this duration helps us determine other elements of motion, such as how far the football will travel horizontally, and how many times it will spin while in the air.
Angular Velocity
Angular velocity is a measure of how quickly an object rotates or spins around an axis. It's expressed in units like revolutions per second (rev/s) or radians per second (rad/s).

In the exercise:
  • We have an angular velocity of \( 7.7 \, \text{rev/s} \), which tells us the ball spins 7.7 times every second.
  • This angular velocity helps keep the spiral motion smooth and consistent, influencing its flight stability.
  • To find the total number of revolutions the ball makes, multiply angular velocity by time of flight: \( 7.7 \, \text{rev/s} \times 3.17 \, \text{s} \approx 24.41 \text{ revolutions} \).
The concept of angular velocity is crucial in sports and many real-world applications where rotational motion needs to be precise and controlled.

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

In the table are listed the initial angular velocity \(\omega_{0}\) and the angular acceleration \(\alpha\) of four rotating objects at a given instant in time. $$ \begin{array}{|c|c|c|} \hline & \text { Initial angular velocity } \omega_{0} & \text { Angular acceleration } \alpha \\ \hline(\mathrm{a}) & +12 \mathrm{rad} / \mathrm{s} & +3.0 \mathrm{rad} / \mathrm{s}^{2} \\ \hline(\mathrm{b}) & +12 \mathrm{rad} / \mathrm{s} & -3.0 \mathrm{rad} / \mathrm{s}^{2} \\ \hline(\mathrm{c}) & -12 \mathrm{rad} / \mathrm{s} & +3.0 \mathrm{rad} / \mathrm{s}^{2} \\ \hline(\mathrm{d}) & -12 \mathrm{rad} / \mathrm{s} & -3.0 \mathrm{rad} / \mathrm{s}^{2} \\ \hline \end{array} $$ In each case, state whether the angular speed of the object is increasing or decreasing in time. Account for your answers. Problem For each of the four pairs in the table, determine the final angular speed of the object if the elapsed time is \(2.0 \mathrm{~s}\). Compare your final angular speeds with the initial angular speeds and make sure that your answers are consistent with your answers to the Concept Question.

One type of slingshot can be made from a length of rope and a leather pocket for holding the stone. The stone can be thrown by whirling it rapidly in a horizontal circle and releasing it at the right moment. Such a slingshot is used to throw a stone from the edge of a cliff, the point of release being \(20.0 \mathrm{~m}\) above the base of the cliff. The stone lands on the ground below the cliff at a point \(X .\) The horizontal distance of point \(X\) from the base of the cliff (directly beneath the point of release) is thirty times the radius of the circle on which the stone is whirled. Determine the angular speed of the stone at the moment of release.

A pitcher throws a curveball that reaches the catcher in \(0.60 \mathrm{~s}\). The ball curves because it is spinning at an average angular velocity of 330 rev \(/ \mathrm{min}\) (assumed constant) on its way to the catcher's mitt. What is the angular displacement of the baseball (in radians) as it travels from the pitcher to the catcher?

The shaft of a pump starts from rest and has an angular acceleration of \(3.00 \mathrm{rad} / \mathrm{s}^{2}\) for \(18.0 \mathrm{~s}\). At the end of this interval, what is (a) the shaft's angular speed and (b) the angle through which the shaft has turned?

At the local swimming hole, a favorite trick is to run horizontally off a cliff that is \(8.3 \mathrm{~m}\) above the water. One diver runs off the edge of the cliff, tucks into a "ball", and rotates on the way down with an average angular speed of 1.6 rev/s. Ignore air resistance and determine the number of revolutions she makes while on the way down.
See all solutions

Recommended explanations on Physics Textbooks

Fundamentals of Physics

Read Explanation

Oscillations

Read Explanation

Engineering Physics

Read Explanation

Electromagnetism

Read Explanation

Scientific Method Physics

Read Explanation

Particle Model of Matter

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.