Momentum

Multiply the mass in kilograms by the velocity in metres per second: p = m·v, with the unit kg·m/s. A football of 0.43 kg at v = 25 m/s has p = 0.43 × 25 ≈ 10.8 kg·m/s. Important: momentum is a vector; it points in the direction of motion, and in problems with oncoming traffic the two directions need opposite signs. Unlike kinetic energy, the velocity enters only linearly: a ball twice as fast has twice the momentum but four times the energy. Convert km/h to m/s before substituting (divide by 3.6), otherwise the unit is wrong.

Both describe motion, but differently: momentum p = m·v is a vector with direction and grows linearly with v; kinetic energy E = ½mv² is a scalar without direction and grows quadratically. The difference becomes decisive in collisions: total momentum is conserved in every collision, kinetic energy only in elastic ones. Two equally heavy carts colliding head-on at equal speed and sticking together have zero total momentum; after the impact both stand still. Their entire kinetic energy then goes into deformation and heat. Mnemonic: momentum describes "impact with direction", energy the "working capacity" of motion.

Impulse is force times interaction time: F·Δt = Δp, equal to the change in momentum. This form of Newton second law explains many safety techniques: the momentum change in an impact is fixed (from v to 0), but the longer the time Δt, the smaller the required force F = Δp/Δt. Airbags, crumple zones and bending your knees on landing extend exactly this time. Worked example: a ball changes its momentum by 10 kg·m/s. With Δt = 0.05 s the force is F = 200 N; if a soft catch stretches the time to 0.5 s, it is only 20 N. In exams this appears as the "average force during impact".

Because velocity has a direction, momentum p = m·v inherits it. In practice this means: in every problem you first fix a positive direction and give motions against it a negative sign. A car with p = +20,000 kg·m/s and an oncoming one with p = −15,000 kg·m/s together have +5,000 kg·m/s, not 35,000. The vector also matters for reversals: if a ball (m = 0.5 kg) hits a wall at 10 m/s and bounces back at 8 m/s, then Δp = 0.5·(−8 − 10) = −9 kg·m/s; the magnitudes add. Anyone who merely subtracts magnitudes here significantly underestimates the momentum change and hence the force.

The SI unit is kilogram metre per second (kg·m/s). It is identical to the newton second (N·s), since 1 N = 1 kg·m/s², so 1 N·s = 1 kg·m/s. This dual form is practical: kg·m/s fits the definition p = m·v, N·s fits the impulse F·Δt = Δp. A quick unit check exposes many arithmetic errors: if you end up with joules (kg·m²/s²), you accidentally used v² and computed energy instead of momentum; if you get newtons, the time is missing in the impulse. Also note: momentum values are only comparable when mass is in kg and velocity in m/s; grams and km/h lead to errors of several orders of magnitude.