How to use this calculator
This tool applies the law of conservation of momentum to a two-body collision. Enter the mass and initial velocity of each object, choose whether the collision is perfectly elastic or perfectly inelastic, and press Calculate to get each object's final velocity plus the total momentum before and after — which should match. Click Clear to reset all fields to the example values.
The formula
Conservation of momentum states that for an isolated system (no external forces), total momentum is the same before and after an interaction:
m1·v1i + m2·v2i = m1·v1f + m2·v2f
Here m1 and m2 are the two masses, v1i and v2i are their velocities before the collision, and v1f and v2f are their velocities afterward. Momentum (p = m·v) is measured in kilogram-meters per second (kg·m/s).
Elastic vs. perfectly inelastic collisions
The equation above only tells you that total momentum is conserved — it takes a second condition to solve for both unknown final velocities. This calculator supports the two standard idealized cases:
- Perfectly elastic: kinetic energy is also conserved, so the objects separate after impact with v1f = ((m1−m2)/(m1+m2))·v1i + (2m2/(m1+m2))·v2i and v2f = (2m1/(m1+m2))·v1i + ((m2−m1)/(m1+m2))·v2i.
- Perfectly inelastic: the objects stick together and share one common final velocity, v_f = (m1·v1i + m2·v2i) / (m1+m2). Kinetic energy is lost to heat, sound, and deformation, but momentum is unaffected.
Understanding the inputs and sign convention
Pick one direction as positive (for example, rightward or "forward") and enter velocities in that same direction consistently; motion the opposite way is a negative number. Masses must be positive and both objects' velocities should use the same unit (m/s). Mixing sign conventions or units between the two objects will produce a meaningless result.
Interpreting the results
The total momentum card shows p = m1v1i + m2v2i, the conserved quantity for this system. The two final-velocity cards show how fast each object moves after the collision — under perfectly inelastic conditions these two values will be identical, since the objects move together. The "momentum after" card recomputes m1v1f + m2v2f as a check: it should equal the total momentum card (aside from rounding), confirming the calculation is internally consistent.
Frequently Asked Questions
What is the law of conservation of momentum?
What is the difference between elastic and perfectly inelastic collisions?
Why does a final velocity come out negative?
Does conservation of momentum apply to explosions and recoil, not just collisions?
Common Tip Percentages
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|---|---|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
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General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|---|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
| Wants | 30% | Dining out, entertainment, hobbies |
| Savings / Debt | 20% | Emergency fund, retirement, extra payments |
Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|---|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |