Momentum

4.3 Momentum

Linear momentum

Because momentum has a direction, a velocity in the opposite direction gives a negative momentum.

Taking the original direction as positive, the momentum goes from $10$ to $-6$ kg m s⁻¹. The change has size 16 kg m s⁻¹. It is not $10 - 6$.

Conservation of momentum

When two objects collide, the forces between them are equal and opposite and last for the same time. So the total momentum does not change, provided no external force acts along the line of motion.

Draw a before-and-after sketch, pick a positive direction, and give any velocity moving the other way a negative sign.

Particles that coalesce

Coalesce means they join together, and move on as one combined particle.

$3 \times 4 + 2 \times 0 = 5v \quad\Rightarrow\quad v = 2.4 \text{ m s}^{-1}$

Particles moving towards each other

Taking $A$'s original direction as positive:

$2(5) + 3(-2) = 2(-1) + 3v \quad\Rightarrow\quad 4 = -2 + 3v \quad\Rightarrow\quad v = 2$

$B$ moves at 2 m s⁻¹ in $A$'s original direction, so $B$'s direction has been reversed too.

Kinetic energy in a collision

Momentum is conserved in these problems, but kinetic energy usually is not: some of it becomes heat and sound.

Before: $\tfrac{1}{2}(2)(5^2) + \tfrac{1}{2}(3)(2^2) = 25 + 6 = 31$ J.

After: $\tfrac{1}{2}(2)(1^2) + \tfrac{1}{2}(3)(2^2) = 1 + 6 = 7$ J.

The loss is $24$ J.

Chains of collisions

Treat each collision on its own, using the velocity after one as the velocity before the next.

First collision: $1 \times 6 = 2v_B$, so $v_B = 3$ m s⁻¹.

Second collision: $2 \times 3 = 4v$, so $v = 1.5$ m s⁻¹.

Joining with other topics

Momentum questions often come with other mechanics:

• Kinematics: find the speed just before the collision using the constant acceleration formulae, and use the speed afterwards for the rest of the motion.
• Energy: find the speed at the bottom of a slope or track, collide, then find how far it travels.
• Friction: the combined particle slows down under friction after impact. Recalculate $R$ and $F$ with the new mass.