Calculate the missing mass or initial/final velocity in a one-dimensional elastic collision using two objects’ masses and velocities.
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Elastic Collision Formula
For a one-dimensional perfectly elastic collision, both momentum and kinetic energy are conserved. The calculator uses the following equations, with all values converted to base units before solving.
v_1 = \frac{m_1 - m_2}{m_1 + m_2}u_1 + \frac{2m_2}{m_1 + m_2}u_2v_2 = \frac{2m_1}{m_1 + m_2}u_1 + \frac{m_2 - m_1}{m_1 + m_2}u_2u_1 = \frac{m_1 - m_2}{m_1 + m_2}v_1 + \frac{2m_2}{m_1 + m_2}v_2u_2 = \frac{2m_1}{m_1 + m_2}v_1 + \frac{m_2 - m_1}{m_1 + m_2}v_2m_1 = \frac{m_2(v_2 - u_2)}{u_1 - v_1}m_2 = \frac{m_1(u_1 - v_1)}{v_2 - u_2}- m1 = mass of object 1
- m2 = mass of object 2
- u1 = initial velocity of object 1
- u2 = initial velocity of object 2
- v1 = final velocity of object 1
- v2 = final velocity of object 2
If you leave final velocity 1 or final velocity 2 blank, the calculator applies the standard one-dimensional elastic collision equations. If you leave an initial velocity blank, it uses the same reversible relationship with initial and final velocities swapped. If you leave a mass blank, it rearranges the momentum relationship to solve for the missing mass.
Common Elastic Collision Patterns
| Situation | Typical result | What it means |
|---|---|---|
| Equal masses, object 2 starts at rest | Object 1 stops, object 2 takes object 1’s speed | The moving object transfers its velocity to the other object. |
| Object 1 is much lighter than object 2 | Object 1 rebounds strongly | A light object bouncing off a heavy object reverses direction. |
| Object 1 is much heavier than object 2 | Object 1 changes little, object 2 moves away quickly | The heavier object keeps most of its motion. |
| Objects move toward each other | One velocity should be entered as negative | Velocity signs show direction in a one-dimensional setup. |
Supported Unit Conversions
| Quantity | Input units | Base unit used for calculation |
|---|---|---|
| Mass | kg, lb | kg |
| Velocity | m/s, ft/s | m/s |
Elastic Collision Examples
Example 1: Find both final velocities for equal masses
Suppose object 1 has a mass of 2 kg and starts at 6 m/s. Object 2 has a mass of 2 kg and starts at 0 m/s.
- m1 = 2 kg
- u1 = 6 m/s
- m2 = 2 kg
- u2 = 0 m/s
For equal masses where the second object starts at rest, the velocities exchange. The final velocity of object 1 is 0 m/s, and the final velocity of object 2 is 6 m/s.
Example 2: Find final velocities for unequal masses
Suppose object 1 has a mass of 3 kg and starts at 4 m/s. Object 2 has a mass of 1 kg and starts at 0 m/s.
v_1 = \frac{3 - 1}{3 + 1}(4) + \frac{2(1)}{3 + 1}(0) = 2 \text{ m/s}v_2 = \frac{2(3)}{3 + 1}(4) + \frac{1 - 3}{3 + 1}(0) = 6 \text{ m/s}The final velocity of object 1 is 2 m/s, and the final velocity of object 2 is 6 m/s.
Elastic Collision FAQ
Can velocity be negative in an elastic collision?
Yes. Velocity is directional. In a one-dimensional collision, choose one direction as positive and enter motion in the opposite direction as negative. For example, if object 1 moves right at 5 m/s and object 2 moves left at 3 m/s, you can enter 5 m/s and -3 m/s.
What makes a collision elastic?
A collision is elastic when total momentum and total kinetic energy are conserved. In real life, many collisions lose some kinetic energy to heat, sound, deformation, or rotation. The calculator assumes a perfectly elastic, one-dimensional collision.
Why does the calculator require exactly one blank field?
The formulas need five known values to solve one missing value. If more than one field is blank, there is not enough information for a single answer. If all six fields are filled, there is nothing left to calculate.
