Roller Coaster Energy Audit at Three Points
A roller coaster car of mass 400 kg (including passengers) starts from rest at point A, 24 m above the ground. It descends to point B at ground level, rises to point C at 15 m, and then enters a vertical loop whose top (point D) is 20 m above the ground. Friction and air resistance remove 6,000 J of mechanical energy between A and B, and another 4,000 J between B and D. A rider insists the car will 'run out of energy' before reaching D because D is higher than C. Assume g = 9.8 m/s². The track is continuous and the car remains on the track throughout.
1. Calculate the car's speed at point B and at point D, accounting for the stated energy losses.
Read the worked answer
At A: v = 0, h = 24 m. E_A = mgh = (400)(9.8)(24) = 94,080 J.
At B (h = 0): E_B = E_A − 6,000 = 94,080 − 6,000 = 88,080 J. All of this is kinetic: ½mv² = 88,080 → v² = 2(88,080)/400 = 440.4 → v ≈ 21.0 m/s.
At D (h = 20 m): E_D = E_A − 6,000 − 4,000 = 84,080 J. Gravitational PE at D: mgh = (400)(9.8)(20) = 78,400 J. Kinetic energy: KE = 84,080 − 78,400 = 5,680 J. ½mv² = 5,680 → v² = 2(5,680)/400 = 28.4 → v ≈ 5.3 m/s.
The car still has positive kinetic energy at D, so it does reach the top of the loop, though slowly.
2. Address the rider's claim. Explain why passing through the lower point C at 15 m does not determine whether the car reaches D at 20 m, and identify what quantity actually governs the outcome.
Read the worked answer
The rider's reasoning treats height alone as the energy budget, but what matters is total mechanical energy relative to the energy required at each height.
At C (h = 15 m), the car has E = 84,080 + 4,000 = 88,080 J (the 4,000 J loss occurs after B, so at C the car still holds the post-B energy minus whatever was lost before C - conservatively at most 88,080 J). Required PE at C is (400)(9.8)(15) = 58,800 J, leaving plenty of KE.
At D, required PE is 78,400 J. The available energy is 84,080 J, so 5,680 J remains as KE. The car reaches D because total energy at D still exceeds the PE needed at that height. The governing quantity is the comparison E_total(at D) versus mgh_D, not the intermediate height C. Since E_D > mgh_D, the car has leftover kinetic energy and does not 'run out.' The rider's claim fails because it ignores that the car carries large kinetic energy through the low points, which is converted back to potential energy on the climb.