Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 3: Problem 57

A \(1 \mathrm{~kg}\) mass traveling in the positive \(x\)-direction at \(3 \mathrm{~m} / \mathrm{s}\) strikes a \(3 \mathrm{~kg}\) mass traveling at \(1 \mathrm{~m} / \mathrm{s}\) in the negative \(x\)-direction. The collision is perfectly elastic. What are possible final velocities for the \(1 \mathrm{~kg}\) mass and the \(3 \mathrm{~kg}\) mass, respectively? (A) \(0 \mathrm{~m} / \mathrm{s} ; 3 \mathrm{~m} / \mathrm{s}\) (B) \(3 \mathrm{~m} / \mathrm{s} ;-1 \mathrm{~m} / \mathrm{s}\) (C) \(-3 \mathrm{~m} / \mathrm{s} ; 1 \mathrm{~m} / \mathrm{s}\) (D) \(3 \mathrm{~m} / \mathrm{s} ; 0 \mathrm{~m} / \mathrm{s}\) (E) \((\mathrm{B})\) and (C)

Short Answer

Expert verified
(A) For mass 1: \(v_{1f} = 0\,\mathrm{m/s}\); For mass 2: \(v_{2f} = 3\,\mathrm{m/s}\) (B) For mass 1: \(v_{1f} = -3\,\mathrm{m/s}\); For mass 2: \(v_{2f} = 1\,\mathrm{m/s}\) (C) Both A and B Ans: (C) Both A and B

Step by step solution

01

Understand the properties of perfectly elastic collision and list the given information

A perfectly elastic collision is a collision in which both the momentum and the kinetic energy are conserved. The given information for this problem is: - Mass 1: \(m_1=1 \mathrm{~kg}\) and initial velocity \(v_{1i}=3 \mathrm{~m} / \mathrm{s}\) - Mass 2: \(m_2=3 \mathrm{~kg}\) and initial velocity \(v_{2i}=-1 \mathrm{~m} / \mathrm{s}\)
02

Define the conservation of momentum and kinetic energy

We know that for any collision, the momentum must be conserved. For this problem, the conservation of momentum can be written as \[ m_1 \cdot v_{1i}+m_2 \cdot v_{2i}= m_1 \cdot v_{1f}+m_2 \cdot v_{2f}\] Both kinetic energy and momentum are conserved for perfectly elastic collision, hence the conservation of kinetic energy can be written as \[ \frac{1}{2} {m_1}{v_{1i}}^2 +\frac{1}{2}{m_2}{v_{2i}}^2 = \frac{1}{2} {m_1}{v_{1f}}^2+\frac{1}{2} {m_2}{v_{2f}}^2\]
03

Plug in the given values and solve for the unknowns

Using the given values, we can rewrite the conservation of momentum equation as \[1 \cdot 3 + 3 \cdot (-1) = 1 \cdot v_{1f}+ 3 \cdot v_{2f}\] And rewrite conservation of kinetic energy as \[\frac{1}{2} \cdot 1 \cdot 3^2 + \frac{1}{2} \cdot 3 \cdot (-1)^2 = \frac{1}{2} \cdot 1 \cdot v_{1f}^2 + \frac{1}{2} \cdot 3 \cdot v_{2f}^2\]
04

Simplify and solve the equations

Simplifying the equations, we get \[ 3 - 3 = v_{1f} + 3v_{2f} \ (1) \] \[ 4.5 + 1.5 = 0.5v_{1f}^2 + 1.5v_{2f}^2 \ (2)\] Now multiply equation (1) by 3 and sum it with equation (2). We obtain: \[6 = v_{1f}^2 + 3v_{2f}^2 \] Now, we can rewrite equation (1) as \(v_{1f} = -3v_{2f}+6\). Substitute this into the new equation: \[ 6 = (-3v_{2f}+6)^2 + 3v_{2f}^2 \] Solving this equation, we obtain the possible values for \(v_{2f}\): \(v_{2f} = 3\,\mathrm{m/s}\) or \(v_{2f} = 1\,\mathrm{m/s}\). Now, we can use equation (1) to find the corresponding values of \(v_{1f}\): - For \(v_{2f} = 3\,\mathrm{m/s}\), we have \(v_{1f} = 0\,\mathrm{m/s}\) - For \(v_{2f} = 1\,\mathrm{m/s}\), we have \(v_{1f} = -3\,\mathrm{m/s}\)
05

Match the solution with the given options

From our calculations, we have found two possible final velocities - For mass 1: \(v_{1f} = 0\,\mathrm{m/s}\) or \(v_{1f} = -3\,\mathrm{m/s}\) - For mass 2: \(v_{2f} = 3\,\mathrm{m/s}\) or \(v_{2f} = 1\,\mathrm{m/s}\) This corresponds to options (A) and (C), so the correct answer is (E) \((\mathrm{B})\) and (C).

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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 crate of mass \(m\) sits atop a frictionless ramp that has a height and length \(L\). The ramp has mass \(2 m\), which is concentrated at the lower left corner, and it sits on a frictionless ice pond. The crate is released and slides down the ramp. By the instant that the crate reaches the bottom, how far has the ramp moved and in what direction? (A) It hasn't moved. (B) It has moved to the left a distance \(L / 3\). (C) It has moved to the right a distance \(L / 3\). (D) It has moved to the left a distance \(L / 2\). (E) It has moved to the right a distance \(L / 2\).

A \(1 \mathrm{~kg}\) mass strikes a \(2 \mathrm{~kg}\) mass in an inelastic collision on a horizontal frictionless surface. The \(1 \mathrm{~kg}\) mass is traveling at \(1 \mathrm{~m} / \mathrm{s}\) in the positive direction and the \(2 \mathrm{~kg}\) mass is traveling at \(1 \mathrm{~m} / \mathrm{s}\) in the negative direction. The final velocity of the masses is (A) \(+1 \mathrm{~m} / \mathrm{s}\) (B) \(+1 / 2 \mathrm{~m} / \mathrm{s}\) (C) \(-1 \mathrm{~m} / \mathrm{s}\) (D) \(-2 / 3 \mathrm{~m} / \mathrm{s}\) (E) \(-1 / 3 \mathrm{~m} / \mathrm{s}\)

An open railroad car is passing at a constant speed \(v_0\) under a water tower, which suddenly dumps a lot of water into the car. The speed of the car will (A) stay the same because of momentum conservation. (B) increase because of momentum conservation. (C) decrease because of momentum conservation. (D) increase because of energy conservation. (E) stay the same because the water is falling in the \(y\)-direction and the car is moving in the \(x\)-direction.

Two identical planets, Alpha and Beta, touch each other. Khan the Conqueror suddenly halves the density of Beta, and doubles the radius of Alpha. The gravitational force between them: (A) stays the same. (B) doubles. (C) halves. (D) goes up by a factor of \(16 / 9\). (E) goes down by a factor of \(9 / 16\).

Your dog Astro is nosing a dinner plate of mass \(M\) across a frozen pond at a constant velocity \(v\). There is a coefficient of friction \(\mu\) between the ice and the block. What is the rate of work Astro does on the plate? (A) \(\mu M v^2\) (B) \(\mu M g v\) (C) \(M g v^2 / \mu\) (D) \(\mu M v^2 / g\) (E) \(\mu M g / v\)
See all solutions

Recommended explanations on Physics Textbooks

Rotational Dynamics

Read Explanation

Scientific Method Physics

Read Explanation

Electric Charge Field and Potential

Read Explanation

Torque and Rotational Motion

Read Explanation

Work Energy and Power

Read Explanation

Quantum Physics

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.

Sign-up for free