Conservation of Momentum Calculator

Conservation of Momentum Calculator solves a two-object collision for final velocities, whether elastic, partially elastic or inelastic. It also finds one unknown mass or velocity.

Using the Conservation of Momentum Calculator

The Conservation of Momentum Calculator treats a collision along one straight line. Enter the mass and starting velocity of each object, then pick the collision type. Motion one way is positive, and motion the other way takes a minus sign.

The first setting, Velocities after the collision, needs only the two masses and the two starting velocities. A perfectly elastic or perfectly inelastic collision is then fully solved. A partially elastic one needs one more fact, either a final velocity or the coefficient of restitution.

The second setting, One unknown value, suits textbook problems where five of the six numbers are given. It returns the missing mass, initial velocity or final velocity. Masses can be in kilograms, grams or pounds, and velocities in m/s, km/h, mph or ft/s.

Every answer comes with a step-by-step table, so the tool doubles as a conservation of momentum calculator with solution. The table shows the momentum before, the solving step, the momentum after and both kinetic energies.

The results assume a closed system. Friction, braking and other outside forces must be small during the short moment of impact, and both objects must move along one line. An angled, two-dimensional collision needs each direction solved on its own.

Conservation of Momentum Formula

Momentum is mass times velocity, and a closed system always conserves it. OpenStax University Physics states that momentum is conserved in every interaction, whether the objects bounce, stick or fly apart. For two objects on a line, the conservation of momentum formula is written below.

$$m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$$

Here m₁ and m₂ are the masses, u₁ and u₂ are the velocities before the collision, and v₁ and v₂ are the velocities after it. One equation cannot give two unknown final velocities, so the collision type supplies the second condition.

Perfectly elastic collision

A perfectly elastic collision also conserves kinetic energy. Solving the two conservation laws together gives the final velocities below. This is the pair the tool uses as an elastic collision conservation of momentum calculator, and it matches the solution printed in OpenStax Example 9.11.

$$v_1 = \frac{(m_1 – m_2) u_1 + 2 m_2 u_2}{m_1 + m_2}$$

$$v_2 = \frac{(m_2 – m_1) u_2 + 2 m_1 u_1}{m_1 + m_2}$$

Perfectly inelastic collision

In a perfectly inelastic collision the objects stick together and share one velocity. The conservation of momentum formula for an inelastic collision then has a single unknown. No other outcome with the same inputs loses more kinetic energy.

$$v = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}$$

Partially elastic collision

Most real impacts sit between those two limits, and the coefficient of restitution, e, says where. MIT’s classical mechanics notes on collision theory define it as the ratio of the relative speed after the collision to the relative speed before it.

$$e = \frac{v_2 – v_1}{u_1 – u_2}$$

A value of 1 is perfectly elastic and 0 is perfectly inelastic. Combining this ratio with the momentum equation gives the momentum after collision formula for any e. The Conservation of Momentum Calculator applies these two lines in its partially elastic setting.

$$v_1 = \frac{m_1 u_1 + m_2 u_2 – m_2 e (u_1 – u_2)}{m_1 + m_2}$$

$$v_2 = \frac{m_1 u_1 + m_2 u_2 + m_1 e (u_1 – u_2)}{m_1 + m_2}$$

Elastic vs Inelastic Collision Results

The table runs one collision three ways in the Conservation of Momentum Calculator. A 5 kg object moving at 3 m/s meets a 2 kg object moving at −1.5 m/s, so they close at 4.5 m/s. Total momentum is 12 kg·m/s before and after in every row.

The kinetic energy card reports the joules lost and the share kept. The loss depends only on the two masses, the closing speed and e, as the next formula shows. With e = 0.5 it gives 10.85 J, the gap between 24.75 J and 13.90 J.

$$\Delta K = \frac{1}{2} \cdot \frac{m_1 m_2}{m_1 + m_2} (1 – e^2) (u_1 – u_2)^2$$

The impulse card is the momentum each object hands to the other. Both receive the same size of impulse in opposite directions, which is Newton’s third law at work. The bouncier the collision, the larger that impulse becomes.

The same card lists the closing speed before and the separation speed after. Their ratio is e, so closing at 4.5 m/s and separating at 2.25 m/s means e = 0.5. The elastic setting shows the center of mass velocity instead, 1.71 m/s here, which is also the shared speed when the objects stick.

Momentum After Collision: Worked Solution

OpenStax Example 9.11 gives a published answer to test the Conservation of Momentum Calculator against. A 12 g hockey puck moving at 2.5 m/s hits a 15 g puck at rest in a perfectly elastic collision. Set both mass units to grams, then enter the 15 g puck as Object 1 at 0 m/s and the 12 g puck as Object 2 at 2.5 m/s.

Momentum before is 0.012 × 2.5 = 0.03 kg·m/s, since the first puck is at rest. The elastic formulas give v₁ = 0.06 ÷ 0.027 = 2.22 m/s and v₂ = −0.0075 ÷ 0.027 = −0.28 m/s. Both values match the textbook result.

The minus sign says the lighter puck bounces back the way it came. Momentum after is 0.015 × 2.22 + 0.012 × (−0.28) = 0.03 kg·m/s, the same as before. Kinetic energy also stays at 0.0375 J, which confirms the elastic label.

A sticking collision works the same way in US units. A 3,000 lb car at 30 mph that locks onto a stopped 4,500 lb SUV leaves both moving at 12 mph. Only 40% of the kinetic energy survives, and the rest goes into bent metal, heat and sound.

Speed before impact matters as much in a crash as it does in the Car Jump Distance Calculator, where it decides how far a car flies off a ramp. Doubling the car’s speed doubles its momentum and doubles the speed of the wreck.

Finding One Unknown Mass or Velocity

Many homework problems give one final velocity and ask for the other. Choose One unknown value, pick the missing quantity and enter the other five. The Conservation of Momentum Calculator rearranges the momentum equation and also reports the coefficient of restitution those numbers imply.

Recoil is the classic case. A 4 kg rifle and a 10 g bullet both start at rest, and the bullet leaves at 800 m/s. Solving for the rifle’s final velocity gives −2 m/s, so the rifle moves backward at 2 m/s.

The result panel labels this an explosion or recoil, because kinetic energy rises from zero to 3,208 J. OpenStax describes the same thing for a fired gun, where momentum is conserved and stored chemical energy becomes kinetic energy. Bullet weight and velocity are also the inputs behind the Ballistic Coefficient Calculator.

An arrow that buries itself in a foam target is the opposite case, a perfectly inelastic collision. Take the arrow’s speed in ft/s from the Arrow Speed Calculator and enter it as Object 1 Initial Velocity. Then set the target’s velocity to 0 and choose the sticking collision type.

Two warnings can appear in this setting. If the entered velocities would need the objects to pass through each other, the result is flagged so the signs can be checked. A mass cannot be found when that object’s velocity does not change, because it took no part in the exchange.

Momentum Calculation for One Object

The momentum of a single object is its mass times its velocity. Mass goes in kilograms and velocity in meters per second, which gives kg·m/s. The table below works through common textbook cases.

$$p = m v$$

The dumbbell row is an NCERT Class 9 exercise. Falling 80 cm with an acceleration of 10 m/s² gives v² = 2 × 10 × 0.8 = 16, so it lands at 4 m/s. All 40 kg·m/s passes to the floor if the dumbbell does not bounce.

Momentum scales directly with mass, so a 35 kg object has 7 times the momentum of a 5 kg object at the same velocity. It scales with velocity too. A 1500 kg car that speeds up from 36 km/h to 72 km/h goes from 10 m/s to 20 m/s, a momentum change of 15,000 kg·m/s.

The Conservation of Momentum Calculator can check any of these rows. Enter the object as Object 1 and set Object 2 Initial Velocity to 0. The Total Momentum card then shows m₁u₁ alone, and the first line of the worked solution lists each object’s own momentum.

Conservation of Momentum Physics FAQs

What is conservation of momentum in simple terms?

The total momentum of a group of objects stays the same as long as no net outside force acts on them. Objects inside the group can trade momentum with each other, but the sum does not change.

When two objects collide, the momentum lost by object 1 equals the momentum gained by object 2. In the calculator’s starting example, object 1 drops from 15 to 2.14 kg·m/s while object 2 rises from −3 to 9.86 kg·m/s. Each changes by 12.86 kg·m/s.

How can I prove the conservation of momentum in Class 9?

The proof uses Newton’s second and third laws. During a collision lasting time t, object 1 pushes on object 2 with a force F, and object 2 pushes back with −F. Each force equals that object’s change in momentum divided by t.

So m₂(v₂ − u₂) ÷ t = −m₁(v₁ − u₁) ÷ t. Cancel t and rearrange to get m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. The impulse card shows those two equal and opposite momentum changes as numbers.

How do you calculate momentum before and after a collision?

Multiply each mass by its velocity, keeping the sign, and add the two results. Do this once with the starting velocities and once with the final ones. For 5 kg at 3 m/s and 2 kg at −1.5 m/s, the total before is 15 + (−3) = 12 kg·m/s.

After an elastic collision the same objects move at 0.43 m/s and 4.93 m/s, which gives 2.14 + 9.86 = 12 kg·m/s again. The worked solution in the conservation of momentum calculator prints both totals, so the two lines can be compared directly.

Can momentum be negative?

Yes. Momentum is a vector, so its sign shows direction along the line of motion. An object moving opposite to the direction chosen as positive has a negative velocity and a negative momentum.

The choice of positive direction is free, but it must be the same for both objects. Two 1000 kg cars meeting head-on at 20 m/s have momenta of +20,000 and −20,000 kg·m/s, which add to zero. If they lock together, the wreck stops where they met.

What are the two formulas for momentum?

The first is p = mv, which gives the momentum of one object. The second is the impulse relation, where a force F acting for a time Δt changes momentum by F × Δt. Rearranged, it shows how to convert momentum to force.

$$F = \frac{\Delta p}{\Delta t}$$

Divide the momentum change by the contact time to get the average force. An impulse of 12.86 N·s delivered in 0.01 s means an average force of about 1,286 N. A longer contact time spreads the same impulse into a smaller force.

Is momentum calculated in kg or g?

The SI unit of momentum is kg·m/s, which is the same as the newton-second. Grams must be converted to kilograms first. A 200 g ball moving at 10 m/s has 0.2 × 10 = 2 kg·m/s of momentum, not 2,000.

The Conservation of Momentum Calculator accepts grams and pounds and converts them internally. With pounds and either ft/s or mph it reports momentum in lb·ft/s, using 1 lb = 0.45359237 kg and 1 ft = 0.3048 m from the NIST conversion factors.

Does it work as a conservation of angular momentum calculator?

No. The Conservation of Momentum Calculator handles linear momentum for objects moving along a straight line. Angular momentum is a separate quantity, L = Iω, where I is the moment of inertia and ω is the angular velocity.

The principle of conservation of angular momentum says L stays constant when no net external torque acts. A spinning skater who pulls both arms in lowers I, so ω rises to keep L the same.