Sure, let's carefully detail the steps required to solve the problem and then fill out the appropriate selections in the drop-down menu.
1. Understanding the Problem:
- Mass of the roller coaster car, \( m \), is 100 kilograms.
- Speed at the top of the hill, \( v_{\text{top}} \), is 3 meters/second.
- Speed at the bottom of the hill, \( v_{\text{bottom}} \), is double the speed at the top. Therefore, \( v_{\text{bottom}} = 2 \times 3 = 6 \) meters/second.
2. Kinetic Energy Calculation at the Top of the Hill:
- The formula for kinetic energy is \( KE = \frac{1}{2} m v^2 \).
- Substituting the values for the top of the hill:
[tex]\[
KE_{\text{top}} = \frac{1}{2} \times 100 \times 3^2 = \frac{1}{2} \times 100 \times 9 = 50 \times 9 = 450 \text{ joules}
\][/tex]
3. Kinetic Energy Calculation at the Bottom of the Hill:
- Using the same kinetic energy formula for the bottom:
[tex]\[
KE_{\text{bottom}} = \frac{1}{2} \times 100 \times 6^2 = \frac{1}{2} \times 100 \times 36 = 50 \times 36 = 1800 \text{ joules}
\][/tex]
4. Filling in the Drop-down Menu:
- Kinetic energy at the bottom of the hill is \( 1800 \text{ joules} \).
- Kinetic energy at the top of the hill is \( 450 \text{ joules} \).
So, the car's kinetic energy at the bottom is [tex]\( \boxed{1800} \)[/tex] joules of kinetic energy at the bottom of the hill. The car has [tex]\( \boxed{450} \)[/tex] joules of kinetic energy at the top.
