To determine the kinetic energy of the kiddie roller coaster car at both the top and the bottom of the hill, we need to use the kinetic energy formula:
[tex]\[ KE = \frac{1}{2} m v^2 \][/tex]
### Step-by-Step Solution:
1. Given Values:
- Mass of the car, [tex]\( m = 100 \)[/tex] kilograms
- Speed at the top of the hill, [tex]\( v_{\text{top}} = 3 \)[/tex] meters/second
2. Kinetic Energy at the Top of the Hill:
[tex]\[ KE_{\text{top}} = \frac{1}{2} \times 100 \times (3)^2 \][/tex]
Simplifying this:
[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]
3. Speed at the Bottom of the Hill:
- The speed at the bottom of the hill is double the speed at the top, so:
[tex]\[ v_{\text{bottom}} = 2 \times 3 = 6 \, \text{meters/second} \][/tex]
4. Kinetic Energy at the Bottom of the Hill:
[tex]\[ KE_{\text{bottom}} = \frac{1}{2} \times 100 \times (6)^2 \][/tex]
Simplifying this:
[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]
5. 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]
### Conclusion:
The kinetic energy of the roller coaster at the bottom of the hill is 4 times its kinetic energy at the top. Thus, the car has 450 joules at the top and 1800 joules at the bottom.
