Momentum, impulse and collisionsAQA A-Level Physics: Revision notes

Linear momentum is the product of mass and velocity:

$p = mv$

It is a vector, measured in kg m s⁻¹ (or N s), with the direction of the velocity. Conservation of linear momentum: the total momentum of a system of interacting bodies in a given direction is constant, provided no external resultant force acts on the system.

For a collision in one dimension, choose a positive direction and write $\sum p_{before} = \sum p_{after}$. Example: a 0.80 kg trolley at 1.5 m s⁻¹ sticks to a stationary 0.40 kg trolley: $0.80\times1.5=(0.80+0.40)v$, so $v=1.0$ m s⁻¹.

In every collision with no external force, momentum is conserved. Kinetic energy may or may not be.

In an elastic collision the total kinetic energy is conserved, and the relative speed of approach equals the relative speed of separation. In an inelastic collision some kinetic energy is transferred to other forms such as thermal energy and sound. Total energy is always conserved.

To test a collision, calculate $E_k=\tfrac12 mv^2$ before and after. For the trolleys: before $0.90$ J, after $0.60$ J, so $0.30$ J is lost and the collision is inelastic.

Newton's second law in its general form: the resultant force equals the rate of change of momentum,

$F=\dfrac{\Delta(mv)}{\Delta t}$

Rearranging for a constant force gives the impulse:

$F\Delta t=\Delta(mv)$

Impulse equals the change in momentum, measured in N s. Example: a 0.43 kg ball leaves a boot at 24 m s⁻¹ after 8.0 ms contact. $\Delta p=0.43\times24=10.3$ N s and $F=10.3\div0.0080=1.3\times10^3$ N.

The area under a force-time graph is the impulse, and therefore the change in momentum. This works even when the force varies with time, so it is used when the force is not constant.

Split the area into triangles and rectangles. A force rising steadily from 0 to 12 N over 3.0 s, then constant at 12 N for 2.0 s gives impulse $\tfrac12\times3.0\times12+12\times2.0=18+24=42$ N s. For a 6.0 kg object starting from rest, $v=42\div6.0=7.0$ m s⁻¹.

The average force is the total impulse divided by the total time.

For a given change in momentum, a longer contact time gives a smaller average force. This is why crumple zones in cars, airbags, seat belts, foam packaging, cycle helmets and the 'give' of a catcher's hands reduce injury or damage.

A car of mass 1100 kg travelling at 14 m s⁻¹ has momentum 15 400 kg m s⁻¹; stopping in 0.15 s needs an average force of $1.0\times10^5$ N, but stopping in 0.025 s needs $6.2\times10^5$ N.

Ethical transport design must consider all road users: in a collision between a light car and a heavy lorry the momentum changes are equal and opposite, so the lighter vehicle has the much larger velocity change and its occupants suffer a far greater deceleration.

In an explosion a stationary system breaks apart; the internal forces are equal and opposite so the total momentum stays zero. The fragments move in opposite directions with equal and opposite momenta, and the total kinetic energy increases because stored energy (chemical or elastic) is transferred to kinetic.

Example: two trolleys of mass 1.0 kg and 2.0 kg are pushed apart by a spring. If the 2.0 kg trolley moves at 0.50 m s⁻¹, then $0=1.0v+2.0(0.50)$, so the 1.0 kg trolley moves at 1.0 m s⁻¹ in the opposite direction. The lighter fragment moves faster.