Momentum:
mass in motion, with a direction

The formula and its units

Two things decide how hard a moving object is to stop: how much of it there is, and how fast it is going. Momentum rolls both into one number. A 2 kg trolley at 5 m/s carries 10 kg·m/s, exactly as much as a 1 kg trolley at 10 m/s.

Unlike speed, momentum is a vector. It points the way the object moves. A ball rolling right and an identical ball rolling left have the same speed but opposite momentum, and that sign is what makes collision sums work.

The unit kg·m/s has no special name in the SI, though it equals the newton-second, which is the unit of impulse. Keep mass in kilograms and speed in metres per second; a car speed of 72 km/h must be divided by 3.6 to give 20 m/s first.

Impulse: changing momentum

Even at the same speed a heavier object has proportionally more momentum, which is why stopping distances for loaded lorries are so much longer than for cars.

Momentum changes only when a force acts for some time. The change is called impulse: J = F × t = Δp. This is Newton's second law in its original form, force equals the rate of change of momentum.

Take a 0.145 kg ball moving at 40 m/s that a catcher brings to rest. The momentum change is 0.145 × 40 = 5.8 kg·m/s. If the catch takes 0.02 s the average force is 5.8 ÷ 0.02 = 290 N. Letting the hand recoil so the catch takes 0.04 s halves the force to 145 N. That is the physics behind padded gloves, crumple zones and bending your knees on landing.

Conservation in collisions

In a closed system with no outside force, total momentum before a collision equals total momentum after. Pick a positive direction, give each object a signed momentum, and add.

If the carts stick together, the 3 kg combined mass must carry the total of +4 kg·m/s, so its speed is 4 ÷ 3 = 1.33 m/s in the positive direction. Kinetic energy is not conserved in a sticky collision, but momentum still is.

Momentum compared with kinetic energy

Momentum scales linearly with speed, while kinetic energy scales with its square. Doubling speed doubles momentum but quadruples energy, so crashes at motorway speeds are far worse than the speed difference suggests.

• Momentum is a vector and is conserved in every isolated collision.
• Kinetic energy is a scalar and is conserved only in perfectly elastic collisions.
• Impulse relates force and time to momentum change; work relates force and distance to energy change.

Momentum in two dimensions

Real collisions rarely happen along a single line. Because momentum is a vector, it is split into components, usually horizontal and vertical, and each component is conserved separately when no outside force acts in that direction.

A ball of mass 0.5 kg moving at 4 m/s east strikes an identical ball at rest, and the two glance off at equal speed on either side of the original line. Before the collision all the momentum, 0.5 × 4 = 2.0 kg·m/s, points east and the north-south total is zero. If each ball leaves at 2.5 m/s, then 2 × 0.5 × 2.5 × cos θ = 2.0, so cos θ = 0.8 and each travels at 36.9° from the original line. The northward and southward components cancel, as they must. If they failed to cancel, the numbers would contradict conservation, which makes this a useful check on a problem or a measurement.

In practice you resolve each velocity with sine and cosine, add components in each direction before and after, and equate them.

Rockets, recoil and everyday cases

Recoil is conservation at work. A 4 kg rifle that fires a 0.01 kg bullet at 600 m/s gives the bullet 6 kg·m/s forward, so the rifle must take 6 kg·m/s backwards, giving a recoil speed of 6 ÷ 4 = 1.5 m/s. A heavier gun recoils more slowly for the same shot, which is why mass is added to firearms and why a rocket works in empty space by throwing mass backwards.

• A swimmer pushing off the wall gains momentum by transferring it from the wall and Earth.
• A car's airbag lengthens the stopping time and lowers peak force for the same momentum change.
• Following through on a golf swing or a tennis stroke keeps the club or racket in contact for longer, which transfers more momentum to the ball.

In each case the same two ideas are in play: the total is conserved in isolation, and force is how fast momentum changes.

Units, magnitudes and sanity checks

Momentum numbers span a huge range, so a feel for scale prevents slips. A walking person carries about 100 kg·m/s, a car on a town road about 15,000 to 30,000, a loaded freight train tens of millions. A tennis serve of 0.057 kg at 50 m/s has just under 3 kg·m/s, and a falling raindrop carries only a few ten-thousandths of a kg·m/s.

A common slip is to confuse momentum with kinetic energy when comparing impacts. A 1,500 kg car at 20 m/s has 30,000 kg·m/s of momentum and ½ × 1,500 × 20² = 300,000 J of kinetic energy. The first tells you the impulse needed to stop it; the second tells you the work the brakes or the crumple zone must absorb. Both matter in a crash, in different ways.

Mistakes worth avoiding

• Dropping the sign when objects move in opposite directions.
• Mixing km/h with kg and expecting kg·m/s.
• Using weight in newtons instead of mass in kilograms.
• Applying conservation of momentum when a significant external force, such as friction with the ground, acts during the collision.