To determine the speed of the roller coaster when it reaches the bottom of the hill, we can use the conservation of energy principle. Energy conservation tells us that the potential energy (PE) lost as the roller coaster falls is converted into kinetic energy (KE).
Here's a step-by-step method to solve the problem:
1. Calculate the potential energy at the top of the hill:
- The potential energy at the top is given by the formula:
[tex]\[
PE_{\text{top}} = m \cdot g \cdot h
\][/tex]
- Where:
- [tex]\( m \)[/tex] is the mass (250 kg),
- [tex]\( g \)[/tex] is the acceleration due to gravity (9.8 m/s²),
- [tex]\( h \)[/tex] is the height (88 m).
2. Substitute the values into the formula:
[tex]\[
PE_{\text{top}} = 250 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 \cdot 88 \, \text{m}
\][/tex]
3. Perform the multiplication:
[tex]\[
PE_{\text{top}} = 250 \cdot 9.8 \cdot 88
\][/tex]
[tex]\[
PE_{\text{top}} = 215,600 \, \text{J} \, (\text{Joules})
\][/tex]
4. Determine the kinetic energy at the bottom of the hill:
- At the bottom of the hill, all the potential energy is converted into kinetic energy.
- Thus:
[tex]\[
KE_{\text{bottom}} = PE_{\text{top}} = 215,600 \, \text{J}
\][/tex]
5. Calculate the speed at the bottom of the hill:
- The kinetic energy at the bottom can be expressed as:
[tex]\[
KE_{\text{bottom}} = \frac{1}{2} m v^2
\][/tex]
- Where [tex]\( v \)[/tex] is the speed at the bottom.
6. Substitute the known values into the kinetic energy formula:
[tex]\[
215,600 = \frac{1}{2} \cdot 250 \cdot v^2
\][/tex]
7. Solve for [tex]\( v^2 \)[/tex]:
[tex]\[
215,600 = 125 v^2
\][/tex]
[tex]\[
v^2 = \frac{215,600}{125}
\][/tex]
[tex]\[
v^2 = 1,724.8
\][/tex]
8. Take the square root to find [tex]\( v \)[/tex]:
[tex]\[
v = \sqrt{1,724.8}
\][/tex]
[tex]\[
v \approx 41.5 \, \text{m/s}
\][/tex]
Thus, the speed of the roller coaster when it reaches the bottom of the hill is approximately [tex]\( 41.5 \, \text{m/s} \)[/tex].
Therefore, the correct answer is:
A. [tex]\( 41.5 \, \text{m/s} \)[/tex]
