Momentum as a Vector and Impulse

Momentum is the product of an object’s mass and velocity, p = mv, a vector quantity measured in kg·m/s, and impulse is the product of a force and the time it acts, J = FΔt, which equals the change in momentum. In brief: momentum points in the direction of the velocity, so it can be positive or negative in one dimension. A heavy slow object and a light fast object can have the same momentum. Impulse is the area under a force-time graph and has the same units as momentum, N·s = kg·m/s.

A moving truck is harder to stop than a bicycle at the same speed, because it has more momentum. Impulse tells how a force acting over a time changes the momentum. This guide defines momentum and impulse, explains their vector nature and units, compares the two quantities in a table, works through calculations including a bat hitting a ball and corrects common mistakes.

What is momentum?

Momentum p = mv measures the “quantity of motion”. Since velocity is a vector, so is momentum, with the same direction. In one dimension, take one direction as positive: an object moving to the left has negative momentum. The SI unit is kg·m/s. Momentum depends on mass and velocity: a 1,000 kg car at 20 m/s has p = 20,000 kg·m/s; a 0.02 kg bullet at 500 m/s has only p = 10 kg·m/s, but because it is stopped over a short distance, it exerts a very large force. Momentum is related to kinetic energy: E_k = p² ÷ 2m. Newton’s second law can be written as F_net = Δp ÷ Δt, so a net force changes the momentum.

What is impulse?

The impulse of a force is J = FΔt for a constant force, in N·s. For a varying force, the impulse is the area under the force-time graph. From F = Δp ÷ Δt it follows that the impulse equals the change in momentum, J = Δp = mv_f − mv_i (the impulse-momentum theorem, covered in more detail in the next article). In collisions, forces are often large and last a short time, so the average force is found as F_avg = Δp ÷ Δt. Both impulse and momentum change are vectors, so reversals of direction count: when a ball bounces back, the change in momentum is larger than the initial momentum.

Momentum and impulse compared

Worked example: calculating momentum and impulse

1. Momentum of different objects. A 1,000 kg car at 20 m/s: p = 20,000 kg·m/s. A 70 kg runner at 8.0 m/s: p = 560 kg·m/s. A 0.020 kg bullet at 500 m/s: p = 10 kg·m/s. A 6.0 × 10²⁴ kg Earth orbiting at 30,000 m/s: p = 1.8 × 10²⁹ kg·m/s.
2. Bat and ball. A 0.145 kg baseball arrives at 40 m/s (take this direction as negative) and leaves the bat at 50 m/s in the opposite direction (positive). Δp = 0.145 × (50 − (−40)) = 0.145 × 90 = 13.05 kg·m/s. The impulse is 13.05 N·s.
3. Average force. If the contact time is 1.0 ms = 0.0010 s, F_avg = 13.05 ÷ 0.0010 = 13,050 N, about 9,000 times the weight of the ball (1.42 N).
4. Graph. A force rises linearly from 0 to 200 N in 0.010 s and then falls to 0 in 0.010 s. The impulse is the area of the triangle: ½ × 0.020 × 200 = 2.0 N·s.

5. Equal momentum. A 4.0 kg object at 5.0 m/s and a 0.50 kg object at 40 m/s each have 20 kg·m/s, but their kinetic energies differ: p² ÷ 2m = 400 ÷ 8 = 50 J, and 400 ÷ 1 = 400 J.

How was momentum defined?

René Descartes proposed in 1644 that “quantity of motion”, the product of the size and speed of a body, is conserved in the universe, but without the vector character. Christiaan Huygens, John Wallis and Christopher Wren corrected the idea in the 1660s by experiments on collisions, showing that direction matters. Newton wrote his second law in terms of change of momentum in the Principia (1687).

Which mistakes are common?

Mistake 1: Ignoring the direction

Momentum is a vector. A bounce reverses the direction, and the change is the sum of the magnitudes.

Mistake 2: Confusing momentum with kinetic energy

Momentum depends on v, kinetic energy on v², and momentum is a vector while energy is a scalar.

Mistake 3: Using the final momentum alone

Impulse is the change in momentum: final minus initial.

Mistake 4: Confusing impulse with force

Impulse is the force multiplied by the time. The same impulse can arise from a large force acting for a short time or a small force acting for a long time.