To solve this, let's break down the problem step-by-step using the formula for kinetic energy, [tex]\( KE = \frac{1}{2} m v^2 \)[/tex].
1. Find the kinetic energy at the top of the hill:
- Mass, [tex]\( m \)[/tex] = 100 kilograms
- Speed at the top, [tex]\( v_{\text{top}} \)[/tex] = 3 meters/second
- Kinetic energy at the top, [tex]\( KE_{\text{top}} \)[/tex]:
[tex]\[
KE_{\text{top}} = \frac{1}{2} \times 100 \, \text{kg} \times (3 \, \text{m/s})^2
\][/tex]
[tex]\[
KE_{\text{top}} = \frac{1}{2} \times 100 \times 9
\][/tex]
[tex]\[
KE_{\text{top}} = 50 \times 9
\][/tex]
[tex]\[
KE_{\text{top}} = 450 \, \text{joules}
\][/tex]
2. Find the kinetic energy at the bottom of the hill:
- Speed at the bottom, [tex]\( v_{\text{bottom}} \)[/tex] = 2 \times 3 meters/second = 6 meters/second
- Kinetic energy at the bottom, [tex]\( KE_{\text{bottom}} \)[/tex]:
[tex]\[
KE_{\text{bottom}} = \frac{1}{2} \times 100 \, \text{kg} \times (6 \, \text{m/s})^2
\][/tex]
[tex]\[
KE_{\text{bottom}} = \frac{1}{2} \times 100 \times 36
\][/tex]
[tex]\[
KE_{\text{bottom}} = 50 \times 36
\][/tex]
[tex]\[
KE_{\text{bottom}} = 1800 \, \text{joules}
\][/tex]
3. Calculate the ratio of kinetic energy at the bottom to the top:
[tex]\[
\text{Ratio} = \frac{KE_{\text{bottom}}}{KE_{\text{top}}}
\][/tex]
[tex]\[
\text{Ratio} = \frac{1800}{450}
\][/tex]
[tex]\[
\text{Ratio} = 4
\][/tex]
So, the kinetic energy of the roller coaster car at the bottom of the hill is 4 times its kinetic energy at the top of the hill.
The car has 450 joules of kinetic energy at the top of the hill and 1800 joules of kinetic energy at the bottom of the hill.
