Momentum Bar Charts

Draw three bars for each object: initial momentum, impulse, and final momentum. The rule tying them together is just the impulse-momentum theorem rearranged: initial momentum plus impulse equals final momentum. Height above the zero line means momentum in the positive direction, height below means momentum in the negative direction. Because the bars are drawn to scale, you can read off an unknown quantity just by seeing how tall its bar has to be to make the stack balance, no algebra required.

For a ball bouncing off a wall, draw the initial bar at the ball's incoming momentum, say +5 kg m/s. The wall pushes back, so the impulse bar points down enough to flip the sign entirely. The final bar lands at -5 kg m/s, same height as the initial bar but below the line. That negative bar is not a smaller bounce, it is the same speed in the reverse direction. Students often want the impulse to be zero since the speed did not change, but speed is not momentum. Momentum is a vector, so reversing direction means the change is twice the original magnitude, never zero.

For a 2 kg cart at 4 m/s hitting and sticking to a 3 kg cart at rest, the bar chart has two initial bars (one per cart), one impulse pair showing equal and opposite impulses on each cart, and one shared final bar since they move together. This setup is the bridge into full collision problems: once you trust the bar chart's bookkeeping, you can write the same statement as an equation, p_initial total equals p_final total, and solve for the common final velocity.