Let's go through the solution step-by-step:
1. Given Data:
- Mass of the roller coaster car, \( m = 100 \) kilograms.
- Speed at the top of the hill, \( v_{\text{top}} = 3 \) meters/second.
- Speed at the bottom of the hill, \( v_{\text{bottom}} = 2 \times v_{\text{top}} = 2 \times 3 = 6 \) meters/second.
2. Kinetic Energy Calculation:
The formula for kinetic energy is given by:
[tex]\[
KE = \frac{1}{2} m v^2
\][/tex]
3. Kinetic Energy at the Top:
Substitute the values \( m = 100 \) kg and \( v_{\text{top}} = 3 \) m/s into the formula:
[tex]\[
KE_{\text{top}} = \frac{1}{2} \times 100 \times (3^2) = \frac{1}{2} \times 100 \times 9 = 450 \text{ joules}
\][/tex]
4. Kinetic Energy at the Bottom:
Substitute the values \( m = 100 \) kg and \( v_{\text{bottom}} = 6 \) m/s into the formula:
[tex]\[
KE_{\text{bottom}} = \frac{1}{2} \times 100 \times (6^2) = \frac{1}{2} \times 100 \times 36 = 1800 \text{ joules}
\][/tex]
5. Ratio of Kinetic Energies:
To find how many times greater the kinetic energy at the bottom is compared to the kinetic energy at the top:
[tex]\[
\text{Ratio} = \frac{KE_{\text{bottom}}}{KE_{\text{top}}} = \frac{1800}{450} = 4
\][/tex]
So, based on the calculations:
- 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.
Let's fill the blanks with the correct answers.
"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."
