Spring Cart Virtual Lab

At the default compression of 0.080 m, k = 10–50 N/m keeps the cart below the ramp top. Compression is adjustable from 0.020 to 0.120 m and changes stored energy ½kx꜀² and entry speed, not the ramp-foot reference point or clock zero. Run trial first shows an illustrative 0.60 s cocking phase, then ideal spring release and a 0.050 m coast to the ramp. Cocking is not a measured physical process. During release the compression follows x꜀ cos(√(k/m)t); release lasts π/(2√(k/m)). The cart reference point at spring release is fixed at x = −0.050 m; the illustrated pusher contacts the rear of the cart. The ramp clock stays “Not started” through setup and starts at ramp entry. The timeline replays only ramp motion. The spring launches the cart horizontally before the ramp. At ramp entry, the speed is v₀ = x꜀√(k/m). This model treats the cart as a point, neglects wheel rotation, friction and drag, and assumes no speed loss at ramp entry. There is no simulated physical apparatus or experimental uncertainty.

While on the ramp: s = v₀t − ½g sin(θ)t² and u = v₀ − g sin(θ)t. Positive u is uphill. The cart is not held at its turning point. When it returns to the foot, it continues left horizontally. Firing is allowed only while the cart is on the ramp. If greater spring energy carries it past the ramp top, it leaves tangent to the ramp, follows ballistic motion, and stops at first ground contact. Firing after departure is rejected; a projectile fired earlier continues normally.

The cannon has an assumed 0.040 m barrel on a fixed hinge. Relative to the cart, the hinge is at (−0.040 cos θ, 0.035 − 0.040 sin θ) m. The muzzle is a further 0.040 m along θ + α. Thus at α = 0° it remains 0.035 m vertically above the cart point; changing α rotates the muzzle around the hinge. Firing does not recoil the cart; the projectile is treated as a negligible-mass test particle. Its separate launch speed w is an adjustable assumption, not energy taken from the cart's propulsion spring. The barrel stays pointed at θ + α relative to horizontal.

At release, xᶠ and yᶠ are the muzzle coordinates from the hinge geometry above. The cart and hinge share velocity u along the ramp. vₓ = u cos(θ) + w cos(θ + α); vᵧ = u sin(θ) + w sin(θ + α). After release: x = xᶠ + vₓτ; y = yᶠ + vᵧτ − ½gτ², where τ is time since release. The flight stops at first contact with the solid ramp or ground, with no bounce. The top of the ramp ends at x = L cos(θ); the ground beyond it remains at y = 0.

The ground target is a deliberately assumed design challenge, centered at x = 0.600 m with half-width 0.025 m. A hit means first ground contact is between x = 0.575 m and 0.625 m. A ramp impact never counts. All positions are relative to the ramp foot. Auto view enlarges the apparatus before firing and fits the flight after firing; axis labels always show meters with the same horizontal and vertical scale. Full flight scale keeps the camera fixed. Cart and projectile icons are enlarged for visibility; the plotted trajectory uses the stated coordinates.