Momentum & Energy

Momentum

p = mv; momentum is a vector quantity.

Change in momentum

Δp = p_f − p_i.

Impulse

J = FΔt = Δp.

Force from impulse

F = Δp/Δt.

Conservation of momentum

Total momentum remains constant in an isolated system.

Why airbags reduce injury

Airbags increase the time over which momentum changes reducing the average force for the same change in momentum.

Kinetic energy

Eₖ = ½mv².

Work done

W = Fd cosθ for a constant force.

Work-energy theorem

The net work done on an object equals its change in kinetic energy: W = ΔEₖ.

Power

P = W/t; power is the rate of energy transfer.

Mechanical energy

The sum of kinetic energy and relevant potential energies.

Conservation of mechanical energy

If only conservative forces do work total mechanical energy remains constant.

Elastic collision

Momentum and kinetic energy are both conserved.

Inelastic collision

Momentum is conserved but kinetic energy is not conserved.

Perfectly inelastic collision

A collision in which objects stick together after colliding.

Gravitational potential energy near Earth

ΔE = mgΔh.

Gravitational potential energy

E = −GMm/r when zero potential energy is defined at infinity.

Elastic potential energy

E = ½kx² for an ideal spring.

Hooke's law

F = −kx; the negative sign indicates that the spring force opposes displacement.

Spring equilibrium

For a vertical spring-mass system at equilibrium the upward spring force balances the downward weight.

Energy transfer in a spring-mass system

Energy can transfer between elastic potential energy gravitational potential energy and kinetic energy.