Question

Discuss the changes in the potential and kinetic energy of a roller-coaster ride as the roller-coaster car climbs to the top and goes down the other side.

Short answer

As a roller-coaster climbs, potential energy increases and kinetic energy decreases. When it descends, potential energy decreases and kinetic energy increases.

Step-by-step solution

- Understanding Potential Energy

Potential energy (PE) is the energy that is stored in an object because of its position relative to a zero potential energy position. In the case of a roller-coaster, the highest point of the track has the maximum potential energy. The formula for potential energy is given by $PE=mgh$, where $m$ is mass, $g$ is the acceleration due to gravity, and $h$ is the height.

- Analyzing Potential Energy at the Top

As the roller-coaster car climbs to the top, its height $h$ increases. Consequently, the potential energy $PE=mgh$ also increases. At the top of the track, the potential energy is at its maximum value.

- Understanding Kinetic Energy

Kinetic energy (KE) is the energy an object possesses due to its motion. The formula for kinetic energy is given by $KE=\frac{1}{2}m{v}^2$, where $m$ is mass and $v$ is the velocity of the object. When the roller-coaster is moving at its fastest, it has the maximum kinetic energy.

- Analyzing Kinetic Energy at the Top

At the top of the hill, the velocity $v$ of the roller-coaster car is very low, and as a result, its kinetic energy $KE=\frac{1}{2}m{v}^2$ is also low. The total energy of the system remains constant, provided there are no external losses such as friction.

- Descending the Hill

As the roller-coaster car goes down the other side, its height $h$ decreases, leading to a decrease in potential energy $PE$. Simultaneously, its velocity $v$ increases, leading to an increase in kinetic energy $KE$.

- Energy Transformation

The key idea is the conservation of energy. The potential energy lost as the car descends is converted into kinetic energy. At the bottom of the hill, the car's kinetic energy is at its maximum while its potential energy is at its minimum.

Key concepts

Potential Energy

Potential energy is the stored energy that an object has due to its position or height. For a roller coaster, when the car climbs up to the top of a hill, it gains potential energy. This is because it is lifted against the force of gravity. The formula to calculate potential energy is: $$PE=mgh$$
where:

• m = mass of the roller coaster car
• g = acceleration due to gravity
• h = height of the car above the ground

As the height (h) increases, the potential energy also increases. The higher the roller coaster climbs, the more potential energy it stores.

Kinetic Energy

Kinetic energy is the energy of motion. When the roller coaster car is moving, it has kinetic energy. The formula for kinetic energy is: $$KE=\frac{1}{2}m{v}^2$$
where:

• m = mass of the roller coaster car
• v = velocity of the car

As the speed (velocity) of the roller coaster car increases, its kinetic energy increases. At the top of the hill, the car has little kinetic energy because its speed is low. As it descends and gains speed, its kinetic energy increases.

Conservation of Energy

The conservation of energy principle states that energy cannot be created or destroyed, only transformed from one form to another. For a roller coaster, this means that the total energy of the system (potential energy + kinetic energy) remains constant, assuming no energy is lost to friction or air resistance. As the roller coaster car goes up the hill, kinetic energy is converted into potential energy. At the highest point, potential energy is at its maximum. As it descends, potential energy transforms back into kinetic energy.
This seamless transformation ensures that the roller coaster continues to move along the track.

Energy Transformation

Energy transformation in roller coasters is an excellent example of how potential and kinetic energy interconvert. When the roller coaster car is at the top of a hill, it has maximum potential energy and minimal kinetic energy because it is almost stationary. As the car begins to descend:

• Potential energy decreases because the height decreases.
• Kinetic energy increases because the car speeds up.

When the car reaches the bottom, it has maximum kinetic energy and minimal potential energy. Throughout the ride, energy is continuously transformed between potential and kinetic forms, demonstrating the law of conservation of energy.