A roller coaster train starts from rest at a height of 47.5 m. What is the speed of the train at 30.0 m below this height?

To determine the speed of the roller coaster train at a height of 30.0 m below its starting height of 47.5 m, we can utilize the principle of conservation of energy, specifically the conversion between gravitational potential energy and kinetic energy.

Initially, at the height of 47.5 m, the train has gravitational potential energy given by the formula:

[ PE = mgh ]

Where:

• ( m ) is the mass of the train,

• ( g ) is the acceleration due to gravity (approximately ( 9.81 , m/s^2 )),

• ( h ) is the height above the reference point.

At the initial height of 47.5 m, the potential energy is:

[ PE_{\text{initial}} = mg \times 47.5 ]

When the train descends to a height of 30.0 m, its height relative to the reference point is reduced. The potential energy at this height is:

[ PE_{\text{final}} = mg \times 30.0 ]

As the train moves down, the potential energy lost will be converted into kinetic energy, which is given by the formula:

[ KE = \frac{1}{2