To determine the kinetic energy of the roller coaster at the top and bottom of the hill, we need to use the kinetic energy formula [tex]\( KE = \frac{1}{2} m v^2 \)[/tex], where [tex]\( m \)[/tex] is the mass and [tex]\( v \)[/tex] is the velocity. Let's break this down step-by-step:
1. Calculate the kinetic energy at the top of the hill:
- Given:
- Mass ([tex]\( m \)[/tex]) = 100 kg
- Speed at the top ([tex]\( v_{\text{top}} \)[/tex]) = 3 m/s
- Use the kinetic energy formula:
[tex]\[ KE_{\text{top}} = \frac{1}{2} \times 100 \, \text{kg} \times (3 \, \text{m/s})^2 \][/tex]
- Simplifying further:
[tex]\[ KE_{\text{top}} = 0.5 \times 100 \times 9 \][/tex]
[tex]\[ KE_{\text{top}} = 450 \, \text{joules} \][/tex]
2. Determine the speed at the bottom of the hill:
- The speed doubles at the bottom, so:
- Speed at the bottom ([tex]\( v_{\text{bottom}} \)[/tex]) = [tex]\( 2 \times 3 \)[/tex] m/s = 6 m/s
3. Calculate the kinetic energy at the bottom of the hill:
- Given:
- Mass ([tex]\( m \)[/tex]) = 100 kg
- Speed at the bottom ([tex]\( v_{\text{bottom}} \)[/tex]) = 6 m/s
- Use the kinetic energy formula:
[tex]\[ KE_{\text{bottom}} = \frac{1}{2} \times 100 \, \text{kg} \times (6 \, \text{m/s})^2 \][/tex]
- Simplifying further:
[tex]\[ KE_{\text{bottom}} = 0.5 \times 100 \times 36 \][/tex]
[tex]\[ KE_{\text{bottom}} = 1800 \, \text{joules} \][/tex]
4. Compare the kinetic energies at the top and bottom:
- The car's kinetic energy at the bottom is [tex]\( \frac{KE_{\text{bottom}}}{KE_{\text{top}}} \)[/tex]:
[tex]\[ \text{Ratio} = \frac{1800 \, \text{joules}}{450 \, \text{joules}} = 4 \][/tex]
So, the car’s kinetic energy at the bottom is 4 times its kinetic energy at the top. The car has 1800 joules of kinetic energy at the bottom of the hill.
The correct answers for the drop-down menu are:
- The car's kinetic energy at the bottom is 4 times its kinetic energy at the top.
- The car has 1800 joules of kinetic energy at the bottom of the hill.
