A roller coaster car tops the first hill carrying 12000 J of gravitational potential energy and 2000 J of kinetic energy.
At the bottom of the dip, its kinetic energy measures 13500 J, and its gravitational potential energy is 0 J.
A frictionless track would deliver the whole account to the bottom as kinetic energy. The car arrived short. How short, and where did that energy go?
Add up the top: 12000 + 2000 = 14000. The car starts with 14000 J of mechanical energy.
Add up the bottom: 13500 + 0 = 13500. Only 13500 J of mechanical energy remains.
14000 - 13500 = 500. The drop in total mechanical energy is 500 J.
The account gains a column for it: E_thermal, the thermal energy that has appeared by that snapshot.
With E_thermal counted, the bottom reads 13500 + 500 = 14000. Nothing is missing; 500 J changed form.
The routine: total the mechanical energy at each snapshot, subtract, and name the difference E_thermal.
Both snapshots need every mechanical store counted: kinetic and gravitational potential at the top, the same pair at the bottom.
A roller coaster car holds 12000 J of gravitational potential energy and 2000 J of kinetic energy at the top of a hill. At the bottom of the dip it holds 13500 J of kinetic energy and 0 J of gravitational potential energy. Calculate the energy dissipated.
Write down the totals at each snapshot:
Top: 12000 + 2000 = 14000. ME = 14000 J.
Bottom: 13500 + 0 = 13500. ME = 13500 J.
Subtract to find the drop:
14000 - 13500 = 500
500 J dissipated: it is thermal energy now, in the wheels, the track, and the air.
A cart crests a hill holding 600 J of gravitational potential energy and 150 J of kinetic energy. At the base, it holds 680 J of kinetic energy and 0 J of gravitational potential energy. Enter the energy dissipated, in J.
A toboggan sets off from rest at the top of a slope, holding 520 J of gravitational potential energy. At the bottom, its kinetic energy is 430 J and its gravitational potential energy is 0 J. Enter the energy dissipated, in J.
A delivery crate leaves the top of a long rough chute holding 2400 J of gravitational potential energy and 100 J of kinetic energy. At the bottom it holds 2100 J of kinetic energy and 0 J of gravitational potential energy. Calculate the energy dissipated.
Write down the totals at each snapshot:
Top: 2400 + 100 = 2500. ME = 2500 J.
Bottom: 2100 + 0 = 2100. ME = 2100 J.
Subtract to find the drop:
2500 - 2100 = 400
400 J dissipated into the crate, the chute, and the surroundings.
A go-kart coasts from the top of a ramp to the bottom. Its total mechanical energy at the bottom is 180 J less than at the top. Gael says: '180 J of thermal energy appeared in the kart, the ramp, and the surroundings.' Is he correct?
A mine cart rolls from the mouth of an incline to the floor below. At the mouth it holds 2400 J of gravitational potential energy and 300 J of kinetic energy. At the floor it holds 2250 J of kinetic energy and 0 J of gravitational potential energy. Enter the energy dissipated on the way down, in J.
A supply sled slides from a ridge down to a bench partway below. At the ridge it holds 1800 J of gravitational potential energy and 200 J of kinetic energy. At the bench it holds 600 J of gravitational potential energy and 1050 J of kinetic energy. Enter the energy dissipated between the two points, in J.
A skateboarder rolls down a bank. At the top she holds 960 J of kinetic energy and 540 J of gravitational potential energy. At the bottom she holds 1210 J of kinetic energy and 0 J of gravitational potential energy. Enter the energy dissipated, in J.