Elastic Collisions

Have you ever played pool or seen someone playing pool? As you play, you hit a ball at a billiard table and that ball hits the other balls on the table, causing them to move as well. A similar situation can be seen while playing volleyball. When you get ready to play, you practice with the ball. When you hit it to the ground, the ball bounces back. These situations are examples of elastic collision. 

Elastic Collisions Elastic Collisions

Create learning materials about Elastic Collisions with our free learning app!

  • Instand access to millions of learning materials
  • Flashcards, notes, mock-exams and more
  • Everything you need to ace your exams
Create a free account
Table of contents

    Elastic Collisions An image of playing poll StudySmarterWhile playing pool, you hit a ball at a billiard table, and that ball hits the other balls. This is an example of elastic collision. Wikimedia Commons

    Elastic Collision Definition

    Elastic collisions are collisions in which the total kinetic energy stays the same before and after the collision. However, the kinetic energies of individual objects can change.

    Elastic Collision: A collision in which the total kinetic energy of the objects stays the same throughout the collision.

    Types of Elastic Collision

    Stationary Target

    As in the example of playing pool, a moving ball can hit a stationary ball and make it move as well. A moving ball has a certain velocity and thus also has kinetic energy. When it hits a stationary ball, it transfers some of its energy to the other ball. This means that the kinetic energy of the moving ball decreases, and the kinetic energy of the stationary ball increases.

    Moving Target

    When both objects are moving and hit each other, an elastic collision can still happen. The total kinetic energy and momentum will still be conserved, but the formulas for the final velocities will differ from the stationary target case.

    Elastic Collisions Formula

    Stationary Target Case

    Elastic Collisions An image of the stationary target case for elastic collision StudySmarterAfter a moving ball hits a stationary ball, it transfers some of its kinetic energy to the other. They both may have a final velocity after the collision. StudySmarter Originals

    When an object with mass \(m_1\) and an initial velocity \(V_{1i}\) hits a stationary ball with mass \(m_2\), the object with mass \(m_1\) has a final velocity \(V_{1f}\) and the object with mass \(m_2\) has a final velocity \(V_{2f}\). The net linear momentum is conserved, that's why the total momentum stays the same before and after the collision.

    The equation for the linear momentum conservation can be written as:

    Equation 1: \(m_1V_{1i}=m_1V_{1f}+m_2V_{2f}\)

    The equation for the kinetic energy conservation can be written as:

    Equation 2: \(\frac 1 2 m_1(V_{1i})^2=\frac 1 2 m_1(V_{1f})^2+\frac 1 2m_2(V_{2f})^2\)

    To find the final velocities, we can do some algebra. We can convert the momentum equation to:

    Equation 3: \(m_1V_{1i}-m_1V_{1f}=m_2V_{2f}\)

    Since \(m_1\) is common on the left side, we can rearrange the equation:

    Equation 4: \(m_1(V_{1i}-V_{1f})=m_2V_{2f}\)

    We can rearrange the energy equation as well. First of all, we can multiply the equation by 2.

    Equation 5: \(m_1(V_{1i})^2=m_1(V_{1f})^2+m_2(V_{2f})^2\)

    We can move \(m_1(V_{1f})^2\) to the left side.

    Equation 6: \(m_1(V_{1i})^2-m_1(V_{1f})^2=m_2(V_{2f})^2\)

    Since \(m_1\) is common on the left side, we can now rearrange the equation:

    Equation 7: \(m_1((V_{1i})^2-(V_{1f})^2)=m_2(V_{2f})^2\)

    Because we are subtracting two items that are in the square form, we can rewrite it as:

    Equation 8: \(m_1(V_{1i}+V_{1f})(V_{1i}-V_{1f})=m_2(V_{2f})^2\)

    After dividing Equation 8 by Equation 4 and then rearranging, we can find the elastic collision formulas for the stationary target case.

    $$V_{1f}=\frac {m_1-m_2}{m_1+m_2}V_{1i}$$

    $$V_{2f}=\frac {2m_1}{m_1+m_2}V_{1i}$$

    Moving Target Case

    Elastic Collisions An image of the moving target case for elastic collision StudySmarterBoth moving balls with velocities can have different final velocities after an elastic collision. The total kinetic energy is still conserved. StudySmarter Originals

    When an object with mass \(m_1\) and an initial velocity \(V_{1i}\) hits a ball with mass \(m_2\) and an initial velocity \(V_{2i}\), the object with mass \(m_1\) has a final velocity \(V_{1f}\) and the object with mass \(m_2\) has a final velocity \(V_{2f}\).

    We can write the momentum conservation equation as:

    Equation 10: \(m_1V_{1i}+m_2V_{2i}=m_1V_{1f}+m_2V_{2f}\)

    We can write the kinetic energy conservation as:

    Equation 11: \(\frac 12m_1(V_{1i})^2+\frac 12m_2(V_{2i})^2=\frac 12m_1(V_{1f})^2+\frac 12m_2(V_{2f})^2\)

    We can rearrange Equation 10:

    Equation 12: \(m_1V_{1i}-m_1V_{1f}=m_2V_{2f}-m_2V_{2i}\)

    Equation 13: \(m_1(V_{1i}-V_{1f})=-m_2(V_{2i}-V_{2f})\)

    Also, we can rearrange Equation 11:

    Equation 14: \(m_1(V_{1i}-V_{1f})(V_{1i}+V_{1f})=-m_2(V_{2i}-V_{2f})(V_{2i}+V_{2f})\)

    We can divide Equation 14 by Equation 13 and rearrange it to get the formulas for the final velocities of the objects:

    $$V_{1f}=\frac{m_1-m_2}{m_1+m_2}V_{1i}+\frac{2m_2}{m_1+m_2}V_{2i}$$

    $$V_{2f}=\frac{2m_1}{m_1+m_2}V_{1i}+\frac{m_2-m_1}{m_1+m_2}V_{2i}$$

    Elastic Collision Equation

    For the stationary target case, our velocity equations are:

    $$V_{1f}=\frac {m_1-m_2}{m_1+m_2}V_{1i}$$

    $$V_{2f}=\frac {2m_1}{m_1+m_2}V_{1i}$$

    For the moving target case, our final velocity equations are different:

    $$V_{1f}=\frac{m_1-m_2}{m_1+m_2}V_{1i}+\frac{2m_2}{m_1+m_2}V_{2i}$$

    $$V_{2f}=\frac{2m_1}{m_1+m_2}V_{1i}+\frac{m_2-m_1}{m_1+m_2}V_{2i}$$

    Elastic Collision Examples

    A ball with a mass of 2 kg is moving to the right with a velocity of 4 m/s and hits a stationary ball with a mass of 1 kg at the billiard table. What are the final velocities of the balls?

    Solution:

    To find the final velocities, we can use Equation 9. As given in the example, \(V_{1i}=4 \frac ms\) and \(m_1=2 kg\). We can insert these values into the equation.

    $$V_{1f}=\frac {2 kg - 1 kg}{2 kg + 1 kg}\times 4 \frac ms = \frac 1 3 \times 4 \frac ms = \frac 4 3 \frac ms$$

    $$V_{2f}=\frac {2 \times 2 kg}{2 kg + 1 kg}\times 4 \frac ms = \frac 4 3 \times 4 \frac ms = \frac {16} 3 \frac ms$$


    A ball with a mass of 6 kg and an initial velocity of 4 m/s hits another ball with a mass of 4 kg and an initial velocity of 2 m/s. They both are moving to the right. What are their final velocities after the collision?

    Solution:

    We can use our formula from above to find the final velocities. Here, \(m_1=6 kg\) and \(m_2=4 kg\), \(V_{1i} = 4 \frac ms\) and \(V_{2i}=2 \frac ms\).

    $$V_{1f}=\frac{6 kg-4 kg}{6 kg+4 kg}\times 4 \frac ms + \frac{2\times 4 kg}{6 kg+4 kg}\times 2 \frac ms$$

    $$V_{1f}= \frac 2{10}\times 4+ \frac 8{10}\times 2 = \frac 8{10}+\frac{16}{10}=\frac{24}{10}\Rightarrow 2.4 \frac ms$$

    $$V_{2f}=\frac{2\times6 kg}{6 kg + 4 kg}\times 4\frac ms + \frac{4 kg - 6 kg}{6 kg + 4 kg}\times 2\frac ms$$

    $$V_{2f} = \frac {12}{10}\times 4 + \frac{-2}{10}\times 2 = \frac{48}{10}-\frac 4{10} = \frac{44}{10}\Rightarrow 4.4 \frac ms$$

    Elastic Collisions - Key takeaways

    • Elastic collisions are collisions in which the total kinetic energy stays the same before and after the collision.
    • The kinetic energies of individual objects can change after the collision.
    • The net linear momentum is conserved.
    • Playing pool and other situations involving bouncing balls can be examples of elastic collision.
    Elastic Collisions Elastic Collisions
    Learn with 18 Elastic Collisions flashcards in the free StudySmarter app

    We have 14,000 flashcards about Dynamic Landscapes.

    Sign up with Email

    Already have an account? Log in

    Frequently Asked Questions about Elastic Collisions

    What are examples of elastic collisions?

    Playing pool and other situations involving bouncing balls can be examples of elastic collision.  

    What is elastic collision?


    Elastic collisions are collisions in which the total kinetic energy stays the same before and after the collision. 


    Is momentum conserved in an elastic collision?


    Yes.

    What is the difference between elastic and inelastic collision?


    While energy is conserved in elastic collisions, it is not conserved in inelastic collisions.

    Is kinetic energy conserved in an elastic collision?


    Yes.

    Discover learning materials with the free StudySmarter app

    Sign up for free
    1
    About StudySmarter

    StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.

    Learn more
    StudySmarter Editorial Team

    Team Physics Teachers

    • 7 minutes reading time
    • Checked by StudySmarter Editorial Team
    Save Explanation

    Study anywhere. Anytime.Across all devices.

    Sign-up for free

    Sign up to highlight and take notes. It’s 100% free.

    Join over 22 million students in learning with our StudySmarter App

    The first learning app that truly has everything you need to ace your exams in one place

    • Flashcards & Quizzes
    • AI Study Assistant
    • Study Planner
    • Mock-Exams
    • Smart Note-Taking
    Join over 22 million students in learning with our StudySmarter App

    Get unlimited access with a free StudySmarter account.

    • Instant access to millions of learning materials.
    • Flashcards, notes, mock-exams, AI tools and more.
    • Everything you need to ace your exams.
    Second Popup Banner