/*! 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 77 The cue ball \(A\) is given an i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 15: Problem 77

The cue ball \(A\) is given an initial velocity \(\left(v_{A}\right)_{1}=5 \mathrm{~m} / \mathrm{s}\). If it makes a direct collision with ball \(B(e=0.8)\), determine the velocity of \(B\) and the angle \(\theta\) just after it rebounds from the cushion at \(C\left(e^{\prime}=0.6\right)\). Each ball has a mass of \(0.4 \mathrm{~kg}\). Neglect their size.

Short Answer

Expert verified
To provide short answer, the systems of equations outlined in the steps above should be solved. It requires numerical computations of the solutions.

Step by step solution

01

Define the system and known variables

Firstly, define the masses of the balls (all equal at 0.4kg), their initial velocities (Ball A starts with 5 m/s, Ball B is stationary at start) and the coefficients of restitution for both collisions (0.8 for A and B, 0.6 for B and C).
02

Collision of Ball A and Ball B

For the first collision between A and B, apply the principle of conservation of linear momentum and equation of the coefficient of restitution. The conservation of linear momentum is given by \( m_A * ((v_A)_1) = m_B * ((v_B)_2) + m_A * ((v_A)_2) \) where \( (v_A)_1 \) is the initial velocity of ball A, and \( (v_A)_2 \) and \( (v_B)_2 \) are the velocities of balls A and B after collision. The equation of the coefficient of restitution, e is \( e = -(v_{B2} - v_{A2})/(v_{A1} - v_{B1}) \) where v_{B1} = 0, thus simplifying to \( e = v_{B2}/v_{A1} \). These equations will form a system of equations that is solvable for \( (v_A)_2 \) and \( (v_B)_2 \).
03

Collision of Ball B and Cushion C

For the second collision where ball B rebounds from cushion C, apply a similar principle. Apply the equation for the coefficient of restitution, \( e' = - (v_{B3} - v_{C2})/v_{B2} \) . Since cushion C is stationary, \(v_{C2} = 0\), we have \( e' = v_{B3}/v_{B2} \) which can be solved for the final velocity of B, \( v_{B3}\). The angle \( \theta \) can be calculated using the equation of motion.
04

Calculate unknowns

Solve the defined systems of equations in steps 2 and 3 respectively, to obtain \( (v_A)_2 \), \( (v_B)_2 \), \( v_{B3} \), and \( \theta \).

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!

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

A ballistic pendulum consists of a \(4-\mathrm{kg}\) wooden block originally at rest, \(\theta=0^{\circ}\). When a 2 -g bullet strikes and becomes embedded in it, it is observed that the block swings upward to a maximum angle of \(\theta=6^{\circ} .\) Estimate the initial speed of the bullet.

A 4-lb ball \(B\) is traveling around in a circle of radius \(r_{1}=3 \mathrm{ft}\) with a speed \(\left(v_{B}\right)_{1}=6 \mathrm{ft} / \mathrm{s}\). If the attached cord is pulled down through the hole with a constant speed \(v_{r}=2 \mathrm{ft} / \mathrm{s}\), determine how much time is required for the ball to reach a speed of \(12 \mathrm{ft} / \mathrm{s}\). How far \(r_{2}\) is the ball from the hole when this occurs? Neglect friction and the size of the ball.

The four smooth balls each have the same mass \(m\). If \(A\) and \(B\) are rolling forward with velocity \(\mathbf{v}\) and strike \(C\), explain why after collision \(C\) and \(D\) each move off with velocity \(\mathbf{v}\). Why doesn't \(D\) move off with velocity \(2 \mathbf{v}\) ? The collision is elastic, \(e=1\). Neglect the size of each ball.

A rocket has an empty weight of \(500 \mathrm{lb}\) and carries 300 lb of fuel. If the fuel is burned at the rate of \(15 \mathrm{lb} / \mathrm{s}\) and ejected with a relative velocity of \(4400 \mathrm{ft} / \mathrm{s}\), determine the maximum speed attained by the rocket starting from rest. Neglect the effect of gravitation on the rocket.

A train consists of a \(50-\mathrm{Mg}\) engine and three cars, each having a mass of \(30 \mathrm{Mg}\). If it takes \(80 \mathrm{~s}\) for the train to increase its speed uniformly to \(40 \mathrm{~km} / \mathrm{h}\), starting from rest, determine the force \(T\) developed at the coupling between the engine \(E\) and the first car \(A\). The wheels of the engine provide a resultant frictional tractive force \(\mathbf{F}\) which gives the train forward motion, whereas the car wheels roll freely. Also, determine \(F\) acting on the engine wheels.
See all solutions

Recommended explanations on Physics Textbooks

Linear Momentum

Read Explanation

Electromagnetism

Read Explanation

Turning Points in Physics

Read Explanation

Force

Read Explanation

Solid State Physics

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.