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.

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
mechanics A 2 kg cart at 3 m/s striking and sticking to a 1 kg cart gives a clean shared 2 m/s.