To find the speed of the roller coaster car at the bottom of the hill, we can use the principle of conservation of energy. Since there is no friction or air resistance, all the gravitational potential energy at the top will convert into kinetic energy at the bottom.

1. Calculate the gravitational potential energy (PE) at the top of the hill: $\text{PE} = m \cdot g \cdot h$ where:

   • $m = 500 \ \text{kg}$ (mass of the car),
   • $g = 9.8 \ \text{m/s}^2$ (acceleration due to gravity),
   • $h = 30 \ \text{m}$ (height of the hill).

   $\text{PE} = 500 \times 9.8 \times 30 = 147,000 \ \text{J}$

2. Set the kinetic energy (KE) at the bottom equal to the potential energy at the top: $\text{KE} = \frac{1}{2} m v^2$ Since all potential energy converts to kinetic energy: $\frac{1}{2} m v^2 = 147,000$

3. Solve for $v$: $v^2 = \frac{2 \times 147,000}{500}$ $v^2 = 588$ $v = \sqrt{588} \approx 24.3 \ \text{m/s}$

Answer: The speed of the car at the bottom of the hill is approximately 24.3 m/s, which matches closest to option:

• B. 24.5 m/s

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Related Questions:

1. What would the speed be if the hill height increased to 50 meters?
2. How would friction affect the speed calculation at the bottom?
3. What is the effect of a change in mass on the final speed in this setup?
4. How does air resistance impact the conversion of potential energy to kinetic energy?
5. How would the result change if gravity was $10 \, \text{m/s}^2$ instead of $9.8 \, \text{m/s}^2$?

Tip:

Conservation of energy is powerful for solving physics problems with no energy losses, as it lets us bypass complex force calculations.