To determine the height difference between the starting point and the lowest point on a track for a roller coaster car, we can use the principle of conservation of mechanical energy. This principle states that the total mechanical energy (potential energy + kinetic energy) in a closed system remains constant if only conservative forces are acting (ignoring friction).
Given:
- Mass of the roller coaster car, [tex]\( m = 440 \)[/tex] kg
- Final velocity at the lowest point, [tex]\( v = 26 \)[/tex] m/s
- Acceleration due to gravity, [tex]\( g = 9.8 \)[/tex] m/s[tex]\(^2\)[/tex]
Step-by-Step Solution:
1. Determine the kinetic energy at the lowest point:
The kinetic energy ([tex]\( KE \)[/tex]) at the lowest point is given by the formula:
[tex]\[
KE = \frac{1}{2}mv^2
\][/tex]
Plugging in the values:
[tex]\[
KE = \frac{1}{2} \times 440 \, \text{kg} \times (26 \, \text{m/s})^2
\][/tex]
Solving this:
[tex]\[
KE = 0.5 \times 440 \times 676
\][/tex]
[tex]\[
KE = 0.5 \times 297440
\][/tex]
[tex]\[
KE = 148720 \, \text{J}
\][/tex]
2. Convert the kinetic energy to potential energy at the top:
At the top of the hill, the roller coaster car starts from rest, so its kinetic energy is zero and all the mechanical energy is in the form of potential energy ([tex]\( PE \)[/tex]). According to the conservation of energy:
[tex]\[
PE_{top} = KE_{bottom}
\][/tex]
The potential energy at a height [tex]\( h \)[/tex] is given by:
[tex]\[
PE = mgh
\][/tex]
Setting this equal to the kinetic energy calculated:
[tex]\[
mgh = 148720 \, \text{J}
\][/tex]
3. Solve for the height [tex]\( h \)[/tex]:
[tex]\[
h = \frac{KE}{mg}
\][/tex]
Substituting the given values:
[tex]\[
h = \frac{148720 \, \text{J}}{440 \, \text{kg} \times 9.8 \, \text{m/s}^2}
\][/tex]
[tex]\[
h = \frac{148720}{4312}
\][/tex]
[tex]\[
h \approx 34.49 \, \text{m}
\][/tex]
Therefore, the height difference between the starting point at the top of the hill and the lowest point on the track is approximately 34 meters, making the correct answer:
34 m
