Question

a 1000.0 kg roller coaster is traveling down a slope. at the first point it is 25 m high andtraveling at 4.0 m/s. at the bottom of the slope it is at ground level and traveling at 20.0 m/ is the amount of energy lost to heat?

Answer 1

Applying the conservation of energy, the roller coaster's initial and final energy components help determine the amount of energy lost to non-conservative forces, likely converted to heat.

To determine the amount of energy lost to heat in the roller coaster scenario, we can apply the conservation of energy principle, which states that the total mechanical energy of an isolated system remains constant if only conservative forces (like gravity) are at play. In the absence of non-conservative forces (like friction), the initial potential energy and kinetic energy should equal the final potential energy and kinetic energy.

The total mechanical energy
`(\(E_{\text{total}}\))` at the initial point
`(height \(h_1\))` is given by the sum of potential energy
`(\(PE_1\))` and kinetic energy
`(\(KE_1\))`:

`\[ E_{\text{total,1}} = PE_1 + KE_1 \]`

Similarly, at the bottom point
`(height \(h_2\))`, the total mechanical energy
`(\(E_{\text{total,2}}\))` is the sum of potential energy
`(\(PE_2\))` and kinetic energy
`(\(KE_2\))`:

`\[ E_{\text{total,2}} = PE_2 + KE_2 \]`

Since the roller coaster starts and ends at ground level, the initial and final potential energies are zero, and the equation simplifies to:

`\[ KE_1 + PE_1 = KE_2 + PE_2 \]`

Now, we can substitute the given values. The potential energy at height (h) is given by
`\(m \cdot g \cdot h\)`, and the kinetic energy is given by
`\((1)/(2) \cdot m \cdot v^2\)`, where (m) is the mass, (g) is the acceleration due to gravity, (h) is the height, and (v) is the velocity.

After substituting and solving, the difference in initial and final mechanical energy gives the amount of energy lost to non-conservative forces, often converted to heat.