Momentum

What momentum measures

• Momentum measures how hard it is to stop a moving object. A slow truck and a fast bullet can be equally difficult to stop.

$$p = mv$$

• $p$ — momentum, measured in kg m s⁻¹ (there is no special name for the unit)

• $m$ — mass (kg)

• $v$ — velocity (m s⁻¹)

• Momentum is a vector: it has the same direction as the velocity.

• A stationary object has zero momentum, whatever its mass.

• Momentum depends on both mass and velocity, so:

  • doubling the mass doubles the momentum,
  • doubling the velocity doubles the momentum.

Momentum is a vector: signs matter

• In one dimension, direction is shown with a sign.
• Choose a positive direction and state it before you calculate.
• An object moving the other way has a negative momentum.
• A $2$ kg ball moving at $3$ m s⁻¹ left, with right as positive, has $p = (2)(-3) = -6$ kg m s⁻¹.

Change in momentum

• The change in momentum is always final minus initial:

$$\Delta p = p_f - p_i = mv_f - mv_i$$

• When an object rebounds, the velocity reverses sign, and the change in momentum is larger than either momentum on its own.
  • A ball hitting a wall at $5$ m s⁻¹ and bouncing back at $5$ m s⁻¹ has $\Delta p = (-5m) - (+5m) = -10m$ — twice what it would be if it simply stopped.
• This is why a bouncing collision exerts a bigger force than one where the object stops dead.

Momentum and kinetic energy are different

Students often blur these. They are not the same quantity:

	Momentum	Kinetic energy
Formula	$p = mv$	$E_k = \tfrac{1}{2}mv^2$
Vector or scalar?	Vector — has direction	Scalar — no direction
Unit	kg m s−1	J
Depends on speed as	$v$	$v^2$
Conserved in a collision?	Always	Only in an elastic collision

• Two objects moving in opposite directions can have a total momentum of zero but a large total kinetic energy — energy has no direction to cancel.