Question
A roller-coaster is at the top of a 62-meter hill. The car and its passengers have a total mass of 1,088 kilograms. By the time the car reaches the bottom of the hill, its speed is 74 miles per hour (33 meters per second). How much kinetic energy does the car have at the bottom of the hill?
KE=12mv2
(1 point)
Responses
2,978,944 J
2,978,944 J
1,184,832 J
1,184,832 J
17,952 J
17,952 J
592,416 J
KE=12mv2
(1 point)
Responses
2,978,944 J
2,978,944 J
1,184,832 J
1,184,832 J
17,952 J
17,952 J
592,416 J
Answers
GPT-4o mini
To find the kinetic energy (KE) of the roller coaster at the bottom of the hill, we can use the formula:
\[
KE = \frac{1}{2} m v^2
\]
where:
- \( m \) is the mass of the car and its passengers (1,088 kilograms).
- \( v \) is the speed at the bottom of the hill (33 meters per second).
Now we can plug in the values:
\[
KE = \frac{1}{2} \times 1088 \, \text{kg} \times (33 \, \text{m/s})^2
\]
First, calculate \( (33 \, \text{m/s})^2 \):
\[
(33 \, \text{m/s})^2 = 1089 \, \text{m}^2/\text{s}^2
\]
Now substitute this back into the equation:
\[
KE = \frac{1}{2} \times 1088 \times 1089
\]
Calculating \( 1088 \times 1089 \):
\[
1088 \times 1089 = 1,180,992
\]
Now, multiply by \( \frac{1}{2} \):
\[
KE = \frac{1}{2} \times 1,180,992 = 590,496 \, \text{J}
\]
However, this value is not listed among the responses. The closest choice is 592,416 J.
Let's double-check:
\[
KE = 6.071 \times 1089 \approx 590,496 \, \text{J}
\]
None of the values given in the multiple-choice answer seem to match the calculated \( KE \) of 590,496 J. There might be an error in calculation or in the listed options. Thus, the answer would be \( 590,496 \, \text{J} \) as per the calculation here.
\[
KE = \frac{1}{2} m v^2
\]
where:
- \( m \) is the mass of the car and its passengers (1,088 kilograms).
- \( v \) is the speed at the bottom of the hill (33 meters per second).
Now we can plug in the values:
\[
KE = \frac{1}{2} \times 1088 \, \text{kg} \times (33 \, \text{m/s})^2
\]
First, calculate \( (33 \, \text{m/s})^2 \):
\[
(33 \, \text{m/s})^2 = 1089 \, \text{m}^2/\text{s}^2
\]
Now substitute this back into the equation:
\[
KE = \frac{1}{2} \times 1088 \times 1089
\]
Calculating \( 1088 \times 1089 \):
\[
1088 \times 1089 = 1,180,992
\]
Now, multiply by \( \frac{1}{2} \):
\[
KE = \frac{1}{2} \times 1,180,992 = 590,496 \, \text{J}
\]
However, this value is not listed among the responses. The closest choice is 592,416 J.
Let's double-check:
\[
KE = 6.071 \times 1089 \approx 590,496 \, \text{J}
\]
None of the values given in the multiple-choice answer seem to match the calculated \( KE \) of 590,496 J. There might be an error in calculation or in the listed options. Thus, the answer would be \( 590,496 \, \text{J} \) as per the calculation here.