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Elastic Collision Calculator

Calculate final velocities in perfectly elastic collisions with momentum and kinetic energy conservation

Parameters

kgⓘ
kgⓘ
m/sⓘ
m/sⓘ
Show Trail

Controls

xⓘ

Calculated Values

Final Velocity 1:
0.33;m/s0.33;m/s
Final Velocity 2:
7.33;m/s7.33;m/s
Initial Momentum:
8.00;kg⋅m/s8.00;kg⋅m/s
Final Momentum:
8.00;kg⋅m/s8.00;kg⋅m/s
Initial Kinetic Energy:
27.00;J27.00;J
Final Kinetic Energy:
27.00;J27.00;J

Examples

Example 1: Equal Mass Collision

Two objects of equal mass (1 kg each) with velocities 5 m/s and -2 m/s.

  • Final Velocity 1: −2.00-2.00
  • Final Velocity 2: 5.005.00
  • Initial Momentum: 3.003.00

Example 2: Heavy Object Hits Light Object

A heavy object (3 kg) moving at 4 m/s hits a light object (1 kg) at rest.

  • Final Velocity 1: 2.002.00
  • Final Velocity 2: 6.006.00
  • Initial Momentum: 12.0012.00

Example 3: Light Object Hits Heavy Object

A light object (1 kg) moving at 6 m/s hits a heavy object (4 kg) at rest.

  • Final Velocity 1: −3.60-3.60
  • Final Velocity 2: 2.402.40
  • Initial Momentum: 6.006.00

Visualization

Elastic Collisions

An elastic collision is a collision in which both momentum and kinetic energy are conserved. The objects bounce off each other without any loss of kinetic energy to other forms of energy such as heat, sound, or deformation.

In elastic collisions, the total momentum of the system before the collision equals the total momentum after the collision. This is a fundamental principle known as the conservation of momentum.

Similarly, the total kinetic energy of the system is conserved. This means that the sum of the kinetic energies of all objects before the collision equals the sum after the collision.

The final velocities in an elastic collision can be calculated using the conservation laws. For two objects moving in one dimension, the final velocities are given by specific formulas that depend on the masses and initial velocities.

Elastic collisions are idealized situations. In real-world collisions, some kinetic energy is always lost to other forms of energy, making them inelastic to some degree. However, many collisions (like those between billiard balls or atoms) are very close to being elastic.

Key Concepts

  • Momentum Conservation: Total momentum before = Total momentum after
  • Kinetic Energy Conservation: Total KE before = Total KE after
  • Elastic Collision: No energy loss to other forms
  • Final Velocities: Calculated using conservation laws
  • One-Dimensional Collision: Objects move along a straight line
  • Perfect Elasticity: Idealized condition with no energy loss

Real-World Applications

  • Particle physics: Collisions between subatomic particles
  • Sports: Billiard ball collisions and ball sports
  • Engineering: Design of impact-absorbing materials
  • Astronomy: Collisions between celestial bodies
  • Molecular dynamics: Atomic and molecular collisions

Explore Further

More mechanics tools

Physics Equations

Final Velocity 1:
v1f=(m1−m2)v1i+2m2v2im1+m2v_{1f} = \frac{(m_1 - m_2)v_{1i} + 2m_2v_{2i}}{m_1 + m_2}
Final Velocity 2:
v2f=(m2−m1)v2i+2m1v1im1+m2v_{2f} = \frac{(m_2 - m_1)v_{2i} + 2m_1v_{1i}}{m_1 + m_2}
Momentum Conservation:
m1v1i+m2v2i=m1v1f+m2v2fm_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}
Kinetic Energy Conservation:
12m1v1i2+12m2v2i2=12m1v1f2+12m2v2f2\frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2

Step-by-Step Solution

See how the main results are calculated.

1

Identify Initial Conditions

List the masses and initial velocities of both objects

Result:

Object1:m1=2kg,v1=5m/sObject2:m2=1kg,v2=−2m/sObject 1: m₁ = 2 kg, v₁ = 5 m/s Object 2: m₂ = 1 kg, v₂ = -2 m/s

Explanation:

We need to identify the masses and initial velocities of both objects before the collision.

2

Calculate Initial Total Momentum

Find the total momentum before the collision

Equation:

pinitial=m1v1+m2v2p_{initial} = m_1v_1 + m_2v_2

Calculation:

pinitial=2×5+1×−2p_{initial} = 2 \times 5 + 1 \times -2
pinitial=10.00+−2.00p_{initial} = 10.00 + -2.00
pinitial=8.00kg⋅m/sp_{initial} = 8.00 \mathrm{kg}\cdot\mathrm{m}/\mathrm{s}

Result:

pinitial=8.00kg⋅m/sp_{initial} = 8.00 \mathrm{kg}\cdot\mathrm{m}/\mathrm{s}

Explanation:

The total momentum is the sum of the momentum of both objects before the collision.

3

Apply Conservation of Momentum

Set initial momentum equal to final momentum

Equation:

pinitial=pfinalp_{initial} = p_{final}

Calculation:

8.00=m1v1′+m2v2′8.00 = m_1v_1' + m_2v_2'
8.00=2v1′+1v2′8.00 = 2v_1' + 1v_2'

Result:

2v1′+1v2′=8.002v_1' + 1v_2' = 8.00

Explanation:

Conservation of momentum states that the total momentum before the collision equals the total momentum after the collision.

4

Elastic Collision Equations

Use elastic collision formulas to find final velocities

Equation:

v1′=(m1−m2)v1+2m2v2m1+m2,v2′=(m2−m1)v2+2m1v1m1+m2v_1' = \frac{(m_1 - m_2)v_1 + 2m_2v_2}{m_1 + m_2}, \quad v_2' = \frac{(m_2 - m_1)v_2 + 2m_1v_1}{m_1 + m_2}

Calculation:

v1′=(2−1)×5+2×1×−22+1v_1' = \frac{(2 - 1) \times 5 + 2 \times 1 \times -2}{2 + 1}
v1′=5.00+−4.003v_1' = \frac{5.00 + -4.00}{3}
v1′=0.33m/sv_1' = 0.33 \mathrm{m}/\mathrm{s}
v2′=(1−2)×−2+2×2×52+1v_2' = \frac{(1 - 2) \times -2 + 2 \times 2 \times 5}{2 + 1}
v2′=2.00+20.003v_2' = \frac{2.00 + 20.00}{3}
v2′=7.33m/sv_2' = 7.33 \mathrm{m}/\mathrm{s}

Result:

v1′=0.33m/s,v2′=7.33m/sv_1' = 0.33 \mathrm{m}/\mathrm{s}, v_2' = 7.33 \mathrm{m}/\mathrm{s}

Explanation:

In elastic collisions, both momentum and kinetic energy are conserved. These equations are derived from both conservation laws.

5

Verify Conservation of Momentum

Check that momentum is conserved

Equation:

pfinal=m1v1′+m2v2′p_{final} = m_1v_1' + m_2v_2'

Calculation:

pfinal=2×0.33+1×7.33p_{final} = 2 \times 0.33 + 1 \times 7.33
pfinal=0.67+7.33p_{final} = 0.67 + 7.33
pfinal=8.00kg⋅m/sp_{final} = 8.00 \mathrm{kg}\cdot\mathrm{m}/\mathrm{s}

Result:

pfinal=8.00kg⋅m/s=pinitialcheckmarkp_{final} = 8.00 \mathrm{kg}\cdot\mathrm{m}/\mathrm{s} = p_{initial} checkmark

Explanation:

The final momentum equals the initial momentum, confirming that momentum is conserved in the collision.

Frequently Asked Questions (FAQ)

What is an elastic collision?

An elastic collision is a collision in which both momentum and kinetic energy are conserved. The objects bounce off each other without any loss of kinetic energy to other forms of energy.

How do you calculate final velocities in an elastic collision?

The final velocities can be calculated using the conservation of momentum and kinetic energy. The formulas are: v₁f = ((m₁-m₂)v₁ᵢ + 2m₂v₂ᵢ)/(m₁+m₂) and v₂f = ((m₂-m₁)v₂ᵢ + 2m₁v₁ᵢ)/(m₁+m₂).

What happens when two objects of equal mass collide elastically?

When two objects of equal mass collide elastically, they exchange velocities. If one object is moving and the other is at rest, the moving object stops and the stationary object moves with the original velocity.

Is momentum always conserved in collisions?

Yes, momentum is always conserved in all types of collisions (elastic and inelastic) as long as there are no external forces acting on the system.

What is the difference between elastic and inelastic collisions?

In elastic collisions, both momentum and kinetic energy are conserved. In inelastic collisions, momentum is conserved but some kinetic energy is lost to other forms of energy like heat, sound, or deformation.

Practice MCQs

  1. In an elastic collision between two objects, which quantity is conserved?
  2. Two objects of equal mass collide elastically. If one is moving at 5 m/s and the other is at rest, what happens?
  3. A heavy object hits a light object elastically. What typically happens to the light object?
  4. What is the total momentum after an elastic collision compared to before?
  5. In a perfectly elastic collision, what happens to the total kinetic energy?

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