Physics › Mechanics › Momentum and impulse
Momentum and impulse
Momentum is the quantity collisions actually trade in, and in a closed system its total is untouchable. Impulse is the same law read over time, and it is why crumple zones, airbags and bent knees work.
Builds on Newton's laws and the resultant force.
IN THIS TOPIC
- Calculate momentum as p = mv and apply conservation of linear momentum to one-dimensional collisions and explosions.
- Use F = Δ(mv)/Δt and impulse FΔt = Δ(mv), including the area under a force-time graph.
- Distinguish elastic from inelastic collisions, and explain contact-time safety design in terms of impulse.
WHAT YOU PROBABLY THINK
When something stops, its momentum simply vanishes.
Momentum, and the general second law
The momentum of an object is
a vector along the velocity, measured in kg m s−1. Newton's second law, stated properly, is about momentum: the resultant force is the rate of change of momentum,
For constant mass this collapses to the familiar F = ma, which is why the shortcut works. The momentum form is the general one, and it keeps working when the mass changes, a rocket burning through its fuel being the classic case.
Conservation
In a closed system, one with no external resultant force, total momentum is conserved: the total before equals the total after, in every collision and every explosion, without exception.
This is Newton's third law doing the accounting. The two colliding bodies exert equal and opposite forces on each other for the same contact time, so their momentum changes are equal and opposite and cancel from the total.
AQA's calculations are one-dimensional, which makes momentum a signed number: choose a positive direction first and let the signs carry the vector work. When a moving object “loses” its momentum by stopping, the momentum has not vanished; it has been transferred, usually to something too large to notice, like the Earth.
Explosions are the same law read backwards. A stationary system has zero total momentum, so after it flies apart the fragments' momenta must still sum to zero: fire a bullet one way and the rifle recoils the other, because zero has to stay zero.
Elastic and inelastic
Momentum is conserved in every collision; kinetic energy is not. An elastic collision conserves kinetic energy too, and is rare outside colliding gas molecules and good billiard balls. An inelastic collision transfers some kinetic energy to internal energy, sound and deformation; a perfectly inelastic one is where the objects stick together.
Say it precisely, because examiners are strict: total energy is always conserved, and that is never in question. Kinetic energy is the thing that need not be. The test is pure arithmetic: compute ½mv2 before and after and compare.
Impulse, and why crumple zones work
Rearranging the second law over a constant force gives the impulse:
The same change in momentum can come from an enormous force acting briefly or a modest force acting longer, and that trade is the whole engineering point.
Crumple zones, airbags, crash mats, bending your knees on landing: none of them change Δ(mv), which is fixed by how fast you were going. They stretch Δt, and because the product is fixed, a longer Δt forces a smaller F. Present the argument in exactly that order. On a force-time graph the area beneath is the impulse, which is how questions handle forces that vary through an impact, where no single F exists. The same reasoning is why vehicle safety design is treated as an ethical obligation as much as a physics exercise: the momentum change in a crash is not negotiable, but the peak force on a human body is.
THE EXAM BIT
- Define a positive direction before writing anything. Most momentum errors are sign errors, and a 1-D collision is a signed-number exercise from the first line.
- “Show that the collision is inelastic” means one thing: compute the total kinetic energy before and after and show it fell. Momentum being conserved is not evidence either way.
- “Energy is lost” loses the mark; “kinetic energy is transferred to internal energy” earns it. Total energy is always conserved.
- The crumple-zone answer runs in fixed order: Δ(mv) is unchanged; the contact time is increased; since F = Δ(mv)/Δt, the force is reduced. Reversing the logic scores nothing.
- On a force-time graph, the area is the impulse, equal to the change in momentum, and it works for any shape of force.
CHECK YOURSELF
A 2.0 kg trolley moving at 3.0 m s−1 collides with a stationary 1.0 kg trolley and they couple together. (a) Find their shared speed. (b) Is the collision elastic?
Show a hint
Momentum settles part (a). Part (b) is a kinetic-energy audit.
Show the answer
(a) Momentum before: 2.0 × 3.0 = 6.0 kg m s−1. After, the same 6.0 is carried by 3.0 kg, so v = 6.0 / 3.0 = 2.0 m s−1.
(b) Kinetic energy before: ½ × 2.0 × 3.02 = 9.0 J. After: ½ × 3.0 × 2.02 = 6.0 J. Kinetic energy fell by 3.0 J, so the collision is inelastic, as coupling collisions always are; the 3.0 J became internal energy and sound.
Momentum is conserved, always.
Kinetic energy is the one you must check.
No animated video for this topic yet; these notes stand alone. InkPhysics on YouTube.