Physics · Mechanics · Grade 10

Momentum and its conservation

How do seat belts and airbags save us?

The situation

At car factories, new cars are deliberately crashed into a concrete wall: inside sits a dummy packed with sensors. Within a tenth of a second the car stops from 50 km/h. How do the seat belt, the airbag and the car's crumple zone save a person? And when two cars collide, where does the momentum go?

Topic

Momentum and its conservation. Crash test: seat belt and airbag

Key ideas

Impulse of a force
Conservation of momentum
Inelastic collision

Formulas

F Δ t = Δ p = m Δ v

Impulse–momentum theorem: for the same $Δ p$ , the longer the stopping time $Δ t$ , the smaller the force.

a = \frac{v_{0}^{2}}{2 s}, Δ t = \frac{2 s}{v_{0}}

Uniform deceleration over the crumple-zone length $s$ : the car's acceleration and stopping time.

m_{1} v_{1} + m_{2} v_{2} = m_{1} v_{1}^{'} + m_{2} v_{2}^{'} = (m_{1} + m_{2}) v_{c}

Law of conservation of momentum: internal forces do not change the total momentum — and therefore do not change the speed $v_{c}$ of the centre of mass either.

e = \frac{v_{2}^{'} - v_{1}^{'}}{v_{1} - v_{2}}

Coefficient of restitution: $e = 1$ — elastic, $e = 0$ — perfectly inelastic collision (the bodies move together).

Δ E = \frac{1}{2} \frac{m_{1} m_{2}}{m_{1} + m_{2}} {(v_{1} - v_{2})}^{2} (1 - e^{2})

Kinetic energy converted into deformation and heat.

Theory

Force × time = change in momentum

To stop the dummy from 50 km/h, its momentum $p = m v_{0}$ must fall to zero in any case — that cannot be changed. What can be changed is the stopping time $Δ t$ : $F = Δ p / Δ t$ . The car's crumple zone lengthens the stopping distance and time. The belt slows the person together with the car along the whole length of the crumple zone, and the airbag gives extra distance and spreads the force over a large area. Acceleration is measured in $g$ : 40 g means a force 40 times greater than the gravitational force.

Dummy without a belt: the “second collision”

Without a belt, as the car decelerates the dummy continues at the previous speed by inertia (Newton's first law). When the car has almost stopped, the dummy hits the steering wheel at a relative speed of ≈ 12 m/s and stops within a few centimetres — the acceleration exceeds 100 g. The airbag is designed to work together with the belt: a modern belt is pulled tight in a collision and the force limiter releases it after 6.5 kN, and the airbag takes up the rest of the distance.

A closed system and conservation of momentum

During the impact external forces such as road friction also act on the cars, but they are tens to hundreds of times smaller than the collision force: the two cars can be taken as a closed system — a set of bodies that interact only with one another. By Newton's third law ${\vec{F}}_{12} = - {\vec{F}}_{21}$ , and by the second law $Δ {\vec{p}}_{1} = {\vec{F}}_{21} Δ t$ and $Δ {\vec{p}}_{2} = {\vec{F}}_{12} Δ t$ — the sum is zero. So the total momentum is conserved in any collision: the $p_{1} + p_{2}$ line on the graph is horizontal. Kinetic energy, however, is conserved only in an elastic collision ( $e = 1$ ); in an inelastic collision part of it turns into deformation and heat — the most when $e = 0$ : if a car hits a stationary car of equal mass, exactly half.

Centre of mass, break-up and conservation laws

The centre of mass (centre of inertia) of a system, $x_{c} = \sum m_{i} x_{i} / \sum m_{i}$ , moves as if the whole mass were concentrated there and only external forces acted on it: $M {\vec{a}}_{c} = {\vec{F}}_{ext}$ . Internal forces do not change it — the “centre of mass” marker (a sphere) on the stage moves uniformly at the speed $v_{c} = p / M$ even during the collision. In the centre-of-mass frame the total momentum is zero: the cars approach and separate with equal and opposite momenta. Break-up is the reverse of a collision: if a body at rest splits in two, the fragments receive equal and opposite momenta, $p = \sqrt{2 μ ε}$ ( $μ = m_{1} m_{2} / (m_{1} + m_{2})$ , $ε$ is the energy released). The recoil of a rifle, jumping from a boat, the rocket (“Jet propulsion and the rocket”) — conservation of momentum in nature and technology. In theoretical mechanics, conservation of momentum is a consequence of the homogeneity of space (Noether's theorem). Conservation laws make it possible to calculate the result without knowing the forces — which is why they are the main tool from car safety to particle physics.

Laboratory: collision of balls

If a ball hung on a string is released from an angle $α$ , at the lowest point it acquires the speed $v = 2 \sqrt{g l} \sin \frac{α}{2}$ (conservation of energy: $v^{2} = 2 g l (1 - \cos α)$ ). From the greatest deflection angles after the collision, $α_{1}^{'}$ and $α_{2}^{'}$ , the speeds $v_{1}^{'}$ and $v_{2}^{'}$ are found by the same formula, and then the momentum, the energy and the coefficient of restitution $e = (v_{2}^{'} - v_{1}^{'}) / v_{1}$ are calculated. The impact is very short and the strings have no time to act — which is why the horizontal momentum is conserved during the impact. For steel balls $e$ is close to one (about 0.9 or more), while balls coated with plasticine stick together and move off as one ( $e = 0$ ).

Model simplifications

The car decelerates uniformly along the crumple zone (a real impact pulse is not uniform). The dummy is a one-dimensional 75 kg body whose chest acceleration is measured; the belt is a spring with 8 cm of slack; in the modern system it is a belt with a pretensioner and a force limiter plus an airbag that inflates in 30 ms; the steering wheel is 45 cm in front of the chest. In the two-car collision the crumpling is a spring with residual deformation; during the short collision time road friction is neglected, and the centre of mass of the cars is taken to be midway between them. The balls are pendulums with inextensible strings, the impact is instantaneous and there is no air resistance. Time is slowed 40 times in the wall test and 10 times in the two-car test; the balls run in real time.

Did you know?

The three-point seat belt was invented in 1959 by the Volvo engineer Nils Bohlin. Volvo made this patent freely available to other carmakers.
The airbag inflates fully about 30 milliseconds after the impact — several times faster than the blink of an eye.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Pascal's law and hydraulics — Grade 7
Elasticity and deformation — Grade 9
Why doesn't a ship sink? — Grade 10
Simple machines: lever, pulley — Grade 10
Basketball: the free throw — Grade 10
Banked turn — Grade 10
Centre of gravity and equilibrium — Grade 10
Aeroplane wing — Grade 10
Dynamics of a rigid body — Grade 10
Non-inertial frames of reference — Grade 10