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Recoil Velocity Formula

Force and laws of motion formulasMomentumClass 9Updated 1 Nov 2026

V=−mvMV = -\frac{mv}{M}
SI unit
m/s
Dimensions
[M⁰ L¹ T⁻¹]
where
  • V recoil velocity of the gun m/s
  • m mass of the bullet kg
  • v velocity of the bullet m/s
  • M mass of the gun kg

The recoil velocity formula is V=−mvMV = -\frac{mv}{M}: when a gun of mass M fires a bullet of mass m at velocity v, the gun moves backward with velocity V. A 4.0 kg rifle firing a 20 g bullet at 400 m/s recoils at 2.0 m/s.

The minus sign means the recoil is opposite to the bullet. The formula comes straight from conservation of momentum: gun and bullet start at rest, so their total momentum is zero before and after firing. The same idea explains a boat drifting back as someone jumps off and a nucleus kicking back as it emits a particle, all part of the laws of motion. The calculator below uses sizes only.

Quick facts

  • Formula

    V=−mvMV = -\frac{mv}{M}

  • From

    0=mv+MV0 = mv + MV

  • SI unit

    m/s

  • Momentum

    Gun and bullet have equal and opposite momentum

  • Kinetic energy ratio

    KgunKbullet=mM\frac{K_{\text{gun}}}{K_{\text{bullet}}} = \frac{m}{M}

  • Heavier gun

    Smaller recoil speed

Deriving the recoil velocity formula

Before firing, the gun and bullet are at rest, so total momentum is zero. The explosion is an internal force, so the total stays zero:

0=mv+MV⇒V=−mvM0 = mv + MV \quad\Rightarrow\quad V = -\frac{mv}{M}

The bullet and the gun receive equal and opposite impulses, which is Newton's third law in action. Since the bullet's mass is tiny compared with the gun's, its speed is large and the gun's speed is small.

Worked examples on recoil

Example 1: a rifle, with grams

A 4.0 kg rifle fires a 20 g bullet at 400 m/s. Find the recoil velocity.

Convert: 20 g = 0.020 kg.

V=−0.020×4004.0=−2.0 m/sV = -\frac{0.020 \times 400}{4.0} = -2.0\ \text{m/s}

The rifle moves back at 2.0 m/s.

Example 2: jumping off a boat

A 60 kg person jumps horizontally off a 140 kg boat at rest, leaving at 2.1 m/s. How fast does the boat move? Ignore water resistance.

V=−60×2.1140=−0.90 m/sV = -\frac{60 \times 2.1}{140} = -0.90\ \text{m/s}

The boat drifts back at 0.90 m/s, which is why jumping from a small boat to a jetty can end in the water.

Example 3: a nucleus emitting an alpha particle

A nucleus at rest emits an alpha particle of mass 4 u at 1.5 × 10⁷ m/s. The remaining nucleus has mass 234 u. Find its recoil speed.

Masses can stay in u, since they appear as a ratio: V=4×1.5×107234V = \frac{4 \times 1.5 \times 10^7}{234}.

V=6.0×107234=2.56×105 m/sV = \frac{6.0 \times 10^7}{234} = 2.56 \times 10^5\ \text{m/s}

Recoil and kinetic energy

Momentum is shared equally, but kinetic energy is not. Since K=p22mK = \frac{p^2}{2m} and both bodies have the same size of momentum, the lighter one takes far more energy. In Example 1, the bullet gets 0.5×0.020×160 000=1600 J0.5 \times 0.020 \times 160\,000 = 1600\ \text{J} and the rifle only 0.5×4.0×4.0=8.0 J0.5 \times 4.0 \times 4.0 = 8.0\ \text{J}. The ratio 8 : 1600 equals m : M = 0.020 : 4.0, as the key facts box says.

That is why the bullet is dangerous and the kick of the gun is merely uncomfortable. See the kinetic energy formula for the energy side.

Reducing the kick

A shooter holds the gun tight against the shoulder. That makes the gun and body move together, so M in the formula becomes the mass of both, and the recoil speed drops. Recoil pads and spring mechanisms do not change the momentum the gun must take, but they lengthen the time over which it is stopped, lowering the force on the shoulder, just as the impulse formula predicts.

Recoil in rockets and jets

A rocket is a gun firing a continuous stream of tiny bullets: gas molecules. Each one carries momentum backward, so the rocket gains momentum forward. Summed over a second, this gives the thrust F=vrΔmΔtF = v_r\frac{\Delta m}{\Delta t}. Sprinklers spinning round and a released balloon zooming across a room follow the same principle.

Recoil mistakes

  • Mixing grams and kilograms in the same ratio.
  • Dropping the minus sign when the question asks for velocity, not speed.
  • Saying the gun has less momentum than the bullet; the sizes are equal.
  • Thinking kinetic energy is shared equally.
If the bullet's speed is given relative to the gun rather than the ground, use v_bullet = v_rel + V. School questions usually give it relative to the ground.

Frequently asked questions

What is the formula for recoil velocity?

V = −mv / M, where m and v are the bullet's mass and velocity and M is the gun's mass. The minus sign shows the gun moves opposite to the bullet.

Why does a gun recoil?

The total momentum of gun and bullet is zero before firing and must stay zero. The bullet gains forward momentum, so the gun gains equal backward momentum.

Does the gun or the bullet have more kinetic energy?

The bullet, by far. Momenta are equal in size, and kinetic energy is p² / 2m, so the lighter body gets more energy.

How can recoil velocity be reduced?

Use a heavier gun or hold it firmly against the body so the effective mass is larger. Padding spreads the stopping over more time, lowering the force felt.