The Cart and the Bumper — Impulse Is the Area
A launcher pushes a dynamics cart along a level track into a spring bumper mounted on a force sensor, while a motion sensor at the other end records the cart's velocity. The cart bounces back and coasts away; press play to run the experiment once, and drag the timeline to any moment of it. Four sliders set the cart's mass m, the launch speed vi, the bumper's stiffness k and its restitution e; they start at 0.50 kg, +0.60 m/s, 600 N/m and 0.80. The velocity and force graphs share one time axis and are drawn up to the moment shown, and every setting you try is logged in a table: vi and vf read from the flat parts of the velocity graph, the contact time Δt, Δp = m(vf − vi) and the area J under the force pulse.
How to use this simulation
- Cart mass m — 0.20 to 1.50 kg, starting at 0.50 kg. The empty cart is 0.20 kg; the steel block on top, drawn to scale, is the rest.
- Launch speed vi — 0.20 to 1.20 m/s, starting at 0.60 m/s.
- Bumper stiffness k — 30 to 1000 N/m on a stretched scale, starting at 600 N/m, with 60 N/m also marked. Each bumper is a 12-turn steel spring drawn to scale: a softer one is longer and wound from thinner wire.
- Bumper restitution e — 0.30 to 1.00, starting at 0.80: the fraction of the speed the bumper gives back. Each new setting is worked out at once and added to the table; nothing starts until you press play.
- Play/pause, back to the start, a timeline you can drag to any moment, and ½×, ¼×, ⅛× slow motion, on top of the 5× slow motion near the bumper.
- Drag anywhere on the scene to look around.
What to look for
- In every row of the table J and Δp agree; at the starting values both are −0.54 N s, whatever the bumper.
- The 60 N/m bumper holds the cart for 0.258 s with a peak of 3.3 N; the 600 N/m bumper holds it for 0.082 s with a peak of 10.4 N. Different shapes, the same area.
- The velocity graph is flat at +0.60 m/s before the collision and at −0.48 m/s after it; during the contact it passes through zero.
- Tripling the mass to 1.50 kg triples Δp to −1.62 N s, while the contact stretches to 0.141 s and the peak rises to 18.0 N — both by a factor of √3.
- Doubling the launch speed to 1.20 m/s doubles Δp to −1.08 N s and the peak to 20.8 N, but the contact still lasts 0.082 s.
- At e = 1.00 the cart leaves at −0.60 m/s and the pulse is symmetric; at e = 0.30 it leaves at only −0.18 m/s and Δp shrinks to −0.39 N s.
The physics behind it
Take the direction towards the bumper as positive. The bumper gives back the fraction e of the speed — it unloads along a steeper line than it loads — so the cart leaves at vf = −e·vi. At the starting values vf = −0.8 × 0.60 = −0.48 m/s and Δp = m(vf − vi) = 0.50 × (−0.48 − 0.60) = −0.54 N s. The force sensor is read 1000 times a second, and the area under its pulse, the impulse J = ∫F dt, comes out at −0.54 N s as well: negative, because the bumper pushes the cart back towards the motion sensor. The stiffness changes only the shape of the pulse. The contact lasts (π/2)(1 + e)√(m/k) and the force peaks at vi√(mk): 0.082 s and 10.4 N at 600 N/m, 0.20 s and 4.2 N at 100 N/m. Longer contact, smaller force, the same area — J = FΔt = Δp. A heavier cart or a faster launch changes Δp, and the area follows it exactly; a bouncier bumper (larger e) changes the momentum more, because Δp = −m·vi(1 + e).