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What is the kinetic energy of the roller coaster at the top and bottom of the hill? Use [tex]$KE = \frac{1}{2} mv^2$[/tex].

A kiddie roller coaster car has a mass of 100 kilograms. At the top of a hill, it is moving at a speed of 3 meters per second. After reaching the bottom of the hill, its speed doubles.

The car's kinetic energy at the bottom is [tex]$\square$[/tex] its kinetic energy at the top. The car has [tex]$\square$[/tex] joules of kinetic energy at the bottom of the hill.



Answer :

To determine the kinetic energy of the roller coaster at the top and bottom of the hill, we need to use the kinetic energy formula [tex]\( KE = \frac{1}{2} m v^2 \)[/tex], where [tex]\( m \)[/tex] is the mass and [tex]\( v \)[/tex] is the velocity. Let's break this down step-by-step:

1. Calculate the kinetic energy at the top of the hill:
- Given:
- Mass ([tex]\( m \)[/tex]) = 100 kg
- Speed at the top ([tex]\( v_{\text{top}} \)[/tex]) = 3 m/s
- Use the kinetic energy formula:
[tex]\[ KE_{\text{top}} = \frac{1}{2} \times 100 \, \text{kg} \times (3 \, \text{m/s})^2 \][/tex]
- Simplifying further:
[tex]\[ KE_{\text{top}} = 0.5 \times 100 \times 9 \][/tex]
[tex]\[ KE_{\text{top}} = 450 \, \text{joules} \][/tex]

2. Determine the speed at the bottom of the hill:
- The speed doubles at the bottom, so:
- Speed at the bottom ([tex]\( v_{\text{bottom}} \)[/tex]) = [tex]\( 2 \times 3 \)[/tex] m/s = 6 m/s

3. Calculate the kinetic energy at the bottom of the hill:
- Given:
- Mass ([tex]\( m \)[/tex]) = 100 kg
- Speed at the bottom ([tex]\( v_{\text{bottom}} \)[/tex]) = 6 m/s
- Use the kinetic energy formula:
[tex]\[ KE_{\text{bottom}} = \frac{1}{2} \times 100 \, \text{kg} \times (6 \, \text{m/s})^2 \][/tex]
- Simplifying further:
[tex]\[ KE_{\text{bottom}} = 0.5 \times 100 \times 36 \][/tex]
[tex]\[ KE_{\text{bottom}} = 1800 \, \text{joules} \][/tex]

4. Compare the kinetic energies at the top and bottom:
- The car's kinetic energy at the bottom is [tex]\( \frac{KE_{\text{bottom}}}{KE_{\text{top}}} \)[/tex]:
[tex]\[ \text{Ratio} = \frac{1800 \, \text{joules}}{450 \, \text{joules}} = 4 \][/tex]

So, 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.

The correct answers for the drop-down menu are:
- 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.