Momentum

Collisions

total momentum is conserved

Colliding carts push equally and oppositely for the same time, so the momentum one loses the other gains; the total is unchanged.

Example

Because colliding carts push on each other equally and oppositely for the same time, the momentum one loses the other gains — the total is unchanged, which pins down the speed after they stick. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Add up the momentum before

Cart A, 2 kilograms at 3 metres per second, has 6 kilogram metres per second. Cart B, 1 kilogram, sits still with zero. The total before is 6.

pbefore=2 kg ⋅ 3 m/s+0=6 kg m/sp_{\text{before}} = 2\ \text{kg}\,\cdot\,3\ \text{m}/\text{s} + 0 = \hl{6}\ \text{kg}\,\text{m}/\text{s}
Before: A moving toward a still BCart A on the left with a velocity arrow moving right toward cart B, which sits still on the right.ABv

Why the total cannot change

During the bump, A pushes B forward and B pushes A back equally hard for the very same time. Equal and opposite forces over the same time are equal and opposite impulses, so whatever momentum A loses, B gains. The total stays put.

pbefore=pafterp_{\text{before}} = p_{\text{after}}

Share the momentum over the joined mass

They stick, so the combined 3 kilograms carries the same 6 kilogram metres per second. The shared speed is 6 over 3, which is 2 metres per second. Momentum is conserved in every collision; sticking is just the easiest because there is a single final speed. Some motion energy is lost in the crunch, though, and the next chapter, energy, follows where it goes.

v=ptotalmtotal=6 kg m/s3 kg=2 m/sv = \frac{p_{\text{total}}}{m_{\text{total}}} = \frac{6\ \text{kg}\,\text{m}/\text{s}}{3\ \text{kg}} = \hl{2}\ \text{m}/\text{s}
After: they stick and move togetherCarts A and B joined together, moving right with a shorter velocity arrow at the shared speed.ABv

More incoming momentum gives more final speed

Keep the joined mass fixed. More total incoming momentum has to be carried after the collision, so the shared speed is larger.

ptotalmtotalv2 kg m/s3 kg23 m/s4 kg m/s3 kg43 m/s6 kg m/s3 kg2 m/s\begin{array}{c|c|c}p_{\text{total}} & m_{\text{total}} & v \\ \hline 2\ \text{kg}\,\text{m}/\text{s} & 3\ \text{kg} & \tfrac{2}{3}\ \text{m}/\text{s} \\ 4\ \text{kg}\,\text{m}/\text{s} & 3\ \text{kg} & \tfrac{4}{3}\ \text{m}/\text{s} \\ 6\ \text{kg}\,\text{m}/\text{s} & 3\ \text{kg} & 2\ \text{m}/\text{s}\end{array}

Same total momentum, more joined mass, less speed

Keep the total momentum fixed. If the same momentum is shared over more joined mass, the final speed is smaller.

ptotalmtotalv6 kg m/s3 kg2 m/s6 kg m/s4 kg32 m/s6 kg m/s6 kg1 m/s\begin{array}{c|c|c}p_{\text{total}} & m_{\text{total}} & v \\ \hline 6\ \text{kg}\,\text{m}/\text{s} & 3\ \text{kg} & 2\ \text{m}/\text{s} \\ 6\ \text{kg}\,\text{m}/\text{s} & 4\ \text{kg} & \tfrac{3}{2}\ \text{m}/\text{s} \\ 6\ \text{kg}\,\text{m}/\text{s} & 6\ \text{kg} & 1\ \text{m}/\text{s}\end{array}
After: they stick and move togetherCarts A and B joined together, moving right with a shorter velocity arrow at the shared speed.ABv
mechanics A 2 kg cart at 3 m/s striking and sticking to a 1 kg cart gives a clean shared 2 m/s.