Let's solve this step-by-step:
1. Convert Height from Feet to Meters:
- The height of the Kingda Ka coaster is 456 feet.
- Convert this height into meters. We know that 1 foot equals 0.3048 meters.
[tex]\[
\text{Height in meters} = 456 \text{ feet} \times 0.3048 \text{ meters/foot} = 138.9888 \text{ meters}
\][/tex]
2. Calculate the Potential Energy at the Top of the Coaster:
- The potential energy (PE) at the top is given by the formula [tex]\(PE = m \cdot g \cdot h\)[/tex], where:
- [tex]\(m\)[/tex] is the mass of the carts (550 kg),
- [tex]\(g\)[/tex] is the acceleration due to gravity (9.81 m/s²),
- [tex]\(h\)[/tex] is the height in meters (138.9888 meters).
[tex]\[
PE_{\text{top}} = 550 \text{ kg} \times 9.81 \text{ m/s}^2 \times 138.9888 \text{ meters} = 749914.0704 \text{ Joules}
\][/tex]
3. Calculate the Kinetic Energy at the Top of the Coaster:
- The kinetic energy (KE) at the top is given by the formula [tex]\(KE = \frac{1}{2} m v^2\)[/tex], where:
- [tex]\(m\)[/tex] is the mass of the carts (550 kg),
- [tex]\(v\)[/tex] is the initial speed (5 m/s).
[tex]\[
KE_{\text{top}} = \frac{1}{2} \times 550 \text{ kg} \times (5 \text{ m/s})^2 = 6875 \text{ Joules}
\][/tex]
4. Total Mechanical Energy at the Top of the Coaster:
- The total energy at the top is the sum of the potential energy and kinetic energy.
[tex]\[
E_{\text{top}} = PE_{\text{top}} + KE_{\text{top}} = 749914.0704 \text{ Joules} + 6875 \text{ Joules} = 756789.0704 \text{ Joules}
\][/tex]
5. At the Bottom of the Coaster:
- When the cart reaches the bottom, the height is 0, hence the potential energy (PE) at the bottom is 0.
- Since energy is conserved, the total mechanical energy at the bottom will be equal to the total mechanical energy at the top, which is [tex]\(756789.0704\)[/tex] Joules.
- Thus, all of the potential energy at the top will convert into kinetic energy at the bottom.
6. Calculate the Final Speed at the Bottom:
- The kinetic energy at the bottom is equal to the total mechanical energy (since potential energy at the bottom is 0):
[tex]\[
KE_{\text{bottom}} = E_{\text{top}} = 756789.0704 \text{ Joules}
\][/tex]
- Use the kinetic energy formula [tex]\(KE = \frac{1}{2} m v^2\)[/tex] and solve for the final speed [tex]\(v\)[/tex]:
[tex]\[
756789.0704 \text{ Joules} = \frac{1}{2} \times 550 \text{ kg} \times v^2
\][/tex]
- Rearranging to solve for [tex]\(v\)[/tex]:
[tex]\[
v^2 = \frac{2 \times 756789.0704 \text{ Joules}}{550 \text{ kg}} \approx 2749.23 \text{ m}^2/\text{s}^2
\][/tex]
[tex]\[
v = \sqrt{2749.23 \text{ m}^2/\text{s}^2} \approx 52.46 \text{ m/s}
\][/tex]
Therefore, the speed of the cart when it reaches the bottom of the roller coaster is approximately 52.46 meters per second.
