Momentum and Newton's second law
Momentum p = mv. The rate of change of momentum equals the net force, which for a body of constant mass gives F = ma.
Why momentum matters
A fast cricket ball is hard to stop, although its mass is small. A slow truck is also hard to stop, because its mass is large. So both mass and velocity decide how a force affects a moving body. Their product is the linear momentum: p = mv. Momentum is a vector. It points the same way as the velocity. It changes if the speed changes, if the direction changes, or both. Its SI unit is kg m s⁻¹.
Derivation: from the second law to F = ma
- 1Second law: the rate of change of momentum of a body is directly proportional to the net external force, and takes place in the direction of the force.
- 2So F ∝ Δp/Δt, or F = k Δp/Δt, where k is a constant.
- 3For a body of constant mass, Δp = m Δv. So F = k m (Δv/Δt).
- 4Δv/Δt is the acceleration a. So F = k m a.
- 5Choose the unit of force so that 1 unit of force gives 1 kg an acceleration of 1 m s⁻². Then 1 = k × 1 × 1, so k = 1.
- 6★So, F = ma. The unit 1 kg m s⁻² is called one newton (N).
★ key step
Points to remember about the second law
- It gives us a way to measure force: by the acceleration it produces in a known mass.
- It is a vector law. The acceleration is along the net force.
- If the net force is zero, a = 0. So the first law is a special case of the second law.
- F = ma holds only when the mass stays constant. A rocket's mass falls as its fuel burns, so its motion is worked out with conservation of momentum instead (see rocket propulsion in the topic on conservation of momentum).
- A 2 kg stone falls freely (g = 9.8 m s⁻²). Its momentum is 0 at the start, 19.6 kg m s⁻¹ after 1 s and 39.2 kg m s⁻¹ after 2 s. Gravity acts along the motion, so the momentum keeps growing.
Worked example: a ball bouncing off a wall
A 0.15 kg ball hits a wall at 12 m s⁻¹ and bounces straight back at 12 m s⁻¹. Find the change in its momentum. If it touches the wall for 0.02 s, find the average force on it.
- Take the direction towards the wall as positive.
- Initial momentum pᵢ = 0.15 kg × 12 m s⁻¹ = +1.8 kg m s⁻¹.
- Final momentum = 0.15 kg × (−12 m s⁻¹) = −1.8 kg m s⁻¹.
- Change Δp = (−1.8) − (+1.8) = −3.6 kg m s⁻¹.
- Average force F = Δp/Δt = −3.6 kg m s⁻¹ ÷ 0.02 s = −180 N.
The momentum changes by 3.6 kg m s⁻¹ away from the wall, and the average force is 180 N away from the wall. The speed is unchanged, but the momentum is not
Worked example: braking force on a car
A car of mass 1200 kg is moving at 20 m s⁻¹. Its brakes bring it to rest in 5 s. Find the braking force, taking it as constant.
- Given: m = 1200 kg, u = 20 m s⁻¹, v = 0, t = 5 s.
- Acceleration a = (v − u)/t = (0 − 20 m s⁻¹) ÷ 5 s = −4 m s⁻².
- Force F = ma = 1200 kg × (−4 m s⁻²) = −4800 N.
- The minus sign shows the force acts opposite to the motion.
A braking force of 4800 N (4.8 kN), opposite to the direction of motion
A fielder or a goalkeeper needs a larger force to stop a faster or heavier ball, because its momentum is larger. Truck drivers keep a long gap on highways for the same reason.
2 marks: “State Newton's second law of motion and define one newton.” The rate of change of momentum of a body is directly proportional to the net external force and takes place in the direction of the force. One newton is the force that gives a 1 kg mass an acceleration of 1 m s⁻².
3 marks: “A 0.5 kg hockey ball moving at 12 m s⁻¹ is brought to rest in 4 s by friction. Find the frictional force, taking it as constant.” Method: a = (v − u)/t, then F = ma. a = (0 − 12) ÷ 4 = −3 m s⁻². F = 0.5 × (−3) = −1.5 N. Answer: 1.5 N, opposite to the motion.
1 marks: “Two bodies have the same momentum. Which one moves faster?” The one with the smaller mass, because v = p/m.
Force is how fast momentum changes: F = Δp/Δt, and with fixed mass, F = ma.