To solve the problem, we'll calculate the potential energy at the top of the hill and then use the conservation of energy principle to find the speed of the roller coaster at the bottom.

1. Potential Energy at the Top of the Hill

The potential energy (PE) can be calculated using the formula: $PE = mgh$ where:

• $m$ = mass (500 kg)
• $g$ = acceleration due to gravity (approximately $9.81 \, \text{m/s}^2$)
• $h$ = height (40 m)

Substituting in the values: $PE = 500 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times 40 \, \text{m}$ $PE = 500 \times 9.81 \times 40 = 196200 \, \text{J}$

2. Speed of the Roller Coaster at the Bottom of the Hill

Using the conservation of mechanical energy, the potential energy at the top will convert to kinetic energy (KE) at the bottom: $KE = \frac{1}{2} mv^2$ Since all potential energy converts to kinetic energy: $PE = KE$ Thus: $196200 \, \text{J} = \frac{1}{2} \times 500 \, \text{kg} \times v^2$

Solving for $v^2$: $v^2 = \frac{2 \times 196200 \, \text{J}}{500 \, \text{kg}} = \frac{392400}{500} = 784.8$

Now, taking the square root to find $v$: $v = \sqrt{784.8} \approx 28.0 \, \text{m/s}$

Summary of Results

• Potential Energy at the Top: $196200 \, \text{J}$
• Speed at the Bottom: $28.0 \, \text{m/s}$

Do you want further details on any part of this solution or have any questions? Here are some related questions you might find interesting:

1. How would the potential energy change if the height were increased to 50 m?
2. What would the speed be at the bottom if friction were taken into account?
3. How does the mass of the roller coaster affect its speed at the bottom?
4. What is the total mechanical energy at the top and bottom of the hill?
5. How can potential energy be calculated for a different height or mass?

Tip: Remember that potential energy is dependent on both height and mass, while kinetic energy depends on mass and speed.