Momentum in two dimensions and collisionsEdexcel International A Level Physics: Revision notes

Momentum is a vector, so in a collision the total momentum is conserved in every direction, provided there is no resultant external force. In two dimensions, resolve each momentum into two perpendicular components (for example $x$ and $y$) and conserve each separately:

$\sum p_x \text{ (before)} = \sum p_x \text{ (after)}$ and $\sum p_y \text{ (before)} = \sum p_y \text{ (after)}$

For a mass $m$ moving at speed $v$ at angle $\theta$ to the $x$ axis, $p_x = mv\cos\theta$ and $p_y = mv\sin\theta$.

In an elastic collision the total kinetic energy is the same before and after. In an inelastic collision some kinetic energy is transferred to thermal energy, sound or deformation, so total kinetic energy decreases.

Momentum is conserved in both types, so to decide, calculate the total kinetic energy before and after and compare. Kinetic energy is a scalar, so add the values for each object. Never resolve kinetic energy into components.

For a non-relativistic particle $E_k = \tfrac12 mv^2$ and $p = mv$, so $v = p/m$. Substituting:

$E_k = \tfrac12 m\left(\dfrac{p}{m}\right)^2 = \dfrac{p^2}{2m}$

This is useful when momentum is known but not speed, and it shows that for the same momentum the lighter object has more kinetic energy.

A 0.16 kg puck moving at 5.0 m s⁻¹ hits an identical stationary puck. The first moves off at 3.0 m s⁻¹ at 53° to its original direction.

• Perpendicular: second puck has $3.0\sin 53^\circ = 2.4$ m s⁻¹ in the opposite sense
• Along line: $5.0 - 3.0\cos 53^\circ = 3.2$ m s⁻¹
• Speed $= \sqrt{3.2^2 + 2.4^2} = 4.0$ m s⁻¹
• $E_k$ before $= 2.0$ J; after $= \tfrac12(0.16)(3.0^2 + 4.0^2) = 2.0$ J, so the collision is elastic.

Roll one ball bearing into a stationary one on a smooth table top and film from above with a scale in view. ICT analysis (video analysis software or a strobe-style stills sequence) gives positions at equal time intervals, so speeds and angles can be found before and after the collision.

1. Calculate the velocities of both spheres before and after.
2. Resolve into components and check that total momentum is conserved.
3. Calculate $E_k$ before and after to decide whether the collision is elastic.

Sources of error include friction, spin and measurement uncertainty in the angles.