Energy Track
Send a ball over hills and through a loop while KE and PE trade off before your eyes.
About this simulation
Drop a ball on a curved track with hills and a loop; live KE / PE / total bars show energy converting as it speeds and climbs. A friction slider bleeds energy away so the bars shrink, and the sim flags whether there is enough speed to complete the loop (v_top² ≥ gR).
How to use it
- Set Release height and watch the ball start from rest on the dashed release line; the KE bar is empty and the PE bar is full.
- Follow the KE and PE bars as the ball runs down into the valley: PE falls, KE rises, and with Friction at 0 the E bar stays the same length.
- Change Loop radius and compare the dashed line marked 'loop needs 2.5R' with the release line; it turns green when the release is high enough and red when it is not.
- Press the Try buttons 'Just clears the loop' and 'Stalls in the loop' to see the ball make it round the top, or leave the track before it (marked with a red ×; the model then slides it back down instead of letting it fly off).
- Raise Friction above 0 and watch the E bar shrink on every pass until the ball comes to rest on the flat.
What to notice
- With no friction, the E bar never changes: every joule of potential energy lost on the way down shows up as kinetic energy, and the reverse on the way up.
- The speed at the floor, √(2gh), does not depend on mass; the 'Mass makes no odds' preset changes the energies but not the motion.
- To stay on the track at the top of the loop the ball needs v² ≥ gR there, which works out to a release height of at least 2.5R, not just 2R.
- Friction converts mechanical energy into heat, so the ball can never climb back to its release height once Friction is above zero.
The formulas
- mgh = ½mv² + mgy (no friction)
- Energy at release height h equals kinetic energy at speed v plus potential energy at the current height y; m is mass, g is 9.8 m/s².
- v = √(2gh)
- Speed at the lowest point of the track after falling from height h; mass cancels out.
- v_top² ≥ gR ⇒ h ≥ 2.5R
- Condition to complete a vertical loop of radius R: the speed at the top must give enough centripetal force with gravity alone.
Questions
What is the law of conservation of mechanical energy?
When only conservative forces such as gravity do work, the sum of kinetic and potential energy stays constant. Energy changes form but the total does not. In the simulation, set Friction to 0 and watch the E bar: it stays the same length while the KE and PE bars swap sizes as the ball rolls down and climbs.
What is the minimum height to complete a vertical loop?
For a body sliding without friction, it must be released from at least 2.5 times the loop radius above the bottom. At the top, gravity must supply the centripetal force, so v² ≥ gR there. Energy conservation then gives 2g(h − 2R) ≥ gR, which simplifies to h ≥ 2.5R. Try a height just above and just below the 'loop needs 2.5R' line; with friction on, that line and the verdict at the top turn red when the run falls off the loop.
Why is h = 2R not enough to loop the loop?
Releasing from 2R would bring the ball to the top of the loop with zero speed, but a ball moving in a circle needs a centripetal force. At zero speed gravity pulls it off the track well before the top. It needs extra height, 0.5R more, so that it still has speed v = √(gR) at the top. The 'Stalls in the loop' preset shows this.
Does a heavier ball go faster down a slope?
No. Both kinetic and potential energy are proportional to mass, so mass cancels from mgh = ½mv² and the speed depends only on the height fallen. A heavier ball has more energy in joules, but reaches the same speed. Change Mass in the simulation: the energy readouts change, the speed labels do not.
Where does the energy go when there is friction?
Work done against friction turns mechanical energy into heat in the ball and the track, so the total kinetic plus potential energy falls. Total energy, including the heat, is still conserved. In the simulation, raise Friction and the E bar gets shorter with every pass until the ball settles at the bottom.
