Use this conservation of momentum calculator to solve 1D two-object collisions for final velocities and to check momentum before vs after, with an optional restitution (e) setting.
Advanced options
How to use our Conservation of Momentum Calculator (1D Collisions)
- In Solve for, choose what you want: solve final velocities from a collision model, or check momentum using final velocities you already know.
- Enter Mass of object 1 (kg) and Mass of object 2 (kg) (both must be greater than 0).
- Enter Initial velocity of object 1 (m/s) and Initial velocity of object 2 (m/s). Pick a positive direction first; anything moving the other way should be negative.
- Open Advanced options and choose a Collision model: Elastic (e = 1), Perfectly inelastic (stick together), or Restitution (0 to 1).
- If you choose Restitution, enter Coefficient of restitution, e (0 to 1). Use 1 for perfectly elastic and 0 for sticking.
- If you chose a check mode, enter both Final velocity of object 1 input (m/s) and Final velocity of object 2 input (m/s) using the same sign convention as your initial velocities.
- (Optional) In Number display and Significant figures (results), choose how results are shown so you do not misread very large or very small numbers.
- Click Calculate.
- Sanity-check: Total momentum before collision, p_before (kg*m/s) should match Total momentum after collision, p_after (kg*m/s), so Momentum check: difference (p_after - p_before) (kg*m/s) should be close to 0 (small differences are usually rounding).
- Use Change in kinetic energy (KE_after - KE_before) (J) to interpret the collision: near 0 suggests elastic; negative means kinetic energy decreased (inelastic).
Definitions
1D collision: A collision where both objects move along one straight line; direction is shown by the sign (+ or -) of velocity.
Momentum (p): A quantity equal to mass times velocity in 1D. It keeps the sign of the velocity, so it also shows direction. In a closed system during the impact, total momentum is conserved. [2]
Total momentum before (p_before) and after (p_after): The sum of each object's momentum before and after the collision; these should match if momentum is conserved. [2]
Elastic collision: A collision where total momentum and total kinetic energy are conserved. [2]
Perfectly inelastic (stick together): After the collision, the objects move together with one shared final velocity; momentum is conserved but kinetic energy decreases. [2]
Coefficient of restitution (e): A number from 0 to 1 that describes how bouncy a 1D collision is (e = 1 perfectly elastic, e = 0 perfectly inelastic). [2]
Kinetic energy (KE): Energy of motion: KE = (1/2) times mass times velocity squared. It is measured in joules (J). [1]
Common mistakes and quick fixes
Mistake: Entering 0 or a negative value for Mass of object 1 (kg) or Mass of object 2 (kg) .
Fix: Use positive mass values only. If your problem gives weight (a force) instead of mass, convert to kg before using this calculator.
Mistake: Forgetting direction and typing speeds instead of signed velocities for Initial velocity of object 1 (m/s) or Initial velocity of object 2 (m/s) .
Fix: Choose one direction as positive, then make motion in the opposite direction negative. If the total momentum before is negative, the system is moving overall in the negative direction.
Mistake: Choosing Restitution (0 to 1) but leaving Coefficient of restitution, e (0 to 1) blank or typing a value outside 0 to 1.
Fix: Enter e between 0 and 1 inclusive. If the objects stick together, use the perfectly inelastic option (or set e = 0).
Mistake: Using Restitution when Initial velocity of object 1 (m/s) equals Initial velocity of object 2 (m/s) and expecting e to change the answer.
Fix: If u1 = u2, the objects have no relative approach speed in 1D, so restitution does not add information; in a closed system the consistent result is v1 = v2 = u1.
Mistake: In a check mode, entering only one of Final velocity of object 1 input (m/s) or Final velocity of object 2 input (m/s) and expecting the calculator to infer the other.
Fix: Enter both final velocities for a momentum check. If you only know one final velocity, switch to a solve mode and provide a collision model (elastic, stick, or e).
Mistake: Thinking Change in kinetic energy (KE_after - KE_before) (J) must be 0 for every collision because momentum is conserved.
Fix: Use the momentum outputs to check conservation of momentum, and use the kinetic energy change to see whether the collision is elastic (near 0) or inelastic (negative).
Methodology
What this calculator does: It models a two-object collision in one dimension (straight line). It solves for final velocity(ies) using your selected collision model, then reports momentum before vs after and the change in kinetic energy so you can check your work. [2]
1) Total momentum before the collision
p_before = (m1 * u1) + (m2 * u2)
Direction comes from the sign of velocity: negative velocity means motion opposite your chosen positive direction.
2) Final velocities, based on collision model
A) Perfectly inelastic (stick together) [2]
V = p_before / (m1 + m2)
Then v1 = V and v2 = V.
B) Perfectly elastic (e = 1) [2]
v1 = ((m1 - m2) / (m1 + m2)) * u1 + (2 * m2 / (m1 + m2)) * u2
v2 = (2 * m1 / (m1 + m2)) * u1 + ((m2 - m1) / (m1 + m2)) * u2
C) Restitution (0 to 1) [2]
v1 = (m1*u1 + m2*u2 - m2*e*(u1 - u2)) / (m1 + m2)
v2 = (m1*u1 + m2*u2 + m1*e*(u1 - u2)) / (m1 + m2)
Special case: if u1 = u2, there is no relative speed to reduce, so restitution does not determine anything new; a consistent outcome is v1 = v2 = u1 (assuming no external impulse). [2]
3) Momentum check
p_after = (m1 * v1) + (m2 * v2)
momentum_error = p_after - p_before
In an ideal closed system this difference is 0. Small nonzero values are usually rounding from displayed results.
4) Kinetic energy before/after [1]
KE_before = 0.5*m1*u1^2 + 0.5*m2*u2^2
KE_after = 0.5*m1*v1^2 + 0.5*m2*v2^2
ke_change = KE_after - KE_before
If ke_change is near 0, the collision is consistent with elastic. If ke_change is negative, kinetic energy decreased (inelastic).
Worked mini-example (restitution)
Given: m1 = 2 kg, m2 = 3 kg, u1 = 4 m/s, u2 = -1 m/s, e = 0.6.
p_before = 2*4 + 3*(-1) = 5 kg*m/s
v1 = (5 - 3*0.6*(4 - (-1))) / 5 = (5 - 9) / 5 = -0.8 m/s
v2 = (5 + 2*0.6*(4 - (-1))) / 5 = (5 + 6) / 5 = 2.2 m/s
p_after = 2*(-0.8) + 3*(2.2) = 5 kg*m/s, so momentum_error = 0
Assumptions and limits
Assumes a short impact where outside forces do not add a meaningful impulse (closed system during the collision) and motion is 1D (head-on along one line). Results can differ in real life if there are external forces during impact, motion is not along one line, objects spin, or energy is added/removed by springs, explosions, or motors. [2]