To solve this problem, we need to find the kinetic energy of the roller coaster car at the top and bottom of the hill.
### Step-by-Step Solution:
1. Determine the given values:
- Mass of the roller coaster car, [tex]\( m \)[/tex] = 100 kg
- Speed at the top of the hill, [tex]\( v_{top} \)[/tex] = 3 m/s
- At the bottom of the hill, the speed doubles, so [tex]\( v_{bottom} = 2 \times v_{top} = 2 \times 3 \)[/tex] m/s = 6 m/s
2. Calculate the kinetic energy at the top of the hill:
The formula for kinetic energy is given by:
[tex]\[
KE = \frac{1}{2} m v^2
\][/tex]
Plugging in the values for the top of the hill:
[tex]\[
KE_{top} = \frac{1}{2} \times 100 \, \text{kg} \times (3 \, \text{m/s})^2
\][/tex]
[tex]\[
KE_{top} = \frac{1}{2} \times 100 \times 9
\][/tex]
[tex]\[
KE_{top} = 450 \, \text{Joules}
\][/tex]
3. Calculate the kinetic energy at the bottom of the hill:
Using the same formula and plugging in the values for the bottom of the hill:
[tex]\[
KE_{bottom} = \frac{1}{2} \times 100 \, \text{kg} \times (6 \, \text{m/s})^2
\][/tex]
[tex]\[
KE_{bottom} = \frac{1}{2} \times 100 \times 36
\][/tex]
[tex]\[
KE_{bottom} = 1800 \, \text{Joules}
\][/tex]
4. Analyze the relationship between the kinetic energies:
To determine how the kinetic energy at the bottom compares to the top, we divide the kinetic energy at the bottom by the kinetic energy at the top:
[tex]\[
\frac{KE_{bottom}}{KE_{top}} = \frac{1800}{450} = 4
\][/tex]
Therefore, the kinetic energy at the bottom is 4 times the kinetic energy at the top.
### Summary:
- 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.
Thus, the correct selections for the drop-down menus are:
1. The car's kinetic energy at the bottom is 4 times its kinetic energy at the top.
2. The car has 1800 Joules of kinetic energy at the bottom of the hill.
