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Answer :
Sure, let's break down the problem step by step.
We are given the potential energy, height, and the acceleration due to gravity. We need to find the mass of the roller coaster.
The formula for potential energy (PE) is:
[tex]\[ PE = m \cdot g \cdot h \][/tex]
Where:
- [tex]\( PE \)[/tex] is the potential energy (in joules),
- [tex]\( m \)[/tex] is the mass (in kilograms) that we need to find,
- [tex]\( g \)[/tex] is the acceleration due to gravity (approximately [tex]\( 9.81 \, \text{m/s}^2 \)[/tex]),
- [tex]\( h \)[/tex] is the height (in meters).
Given:
- [tex]\( PE = 235200 \, \text{J} \)[/tex]
- [tex]\( h = 30 \, \text{m} \)[/tex]
- [tex]\( g = 9.81 \, \text{m/s}^2 \)[/tex]
We need to rearrange the formula to solve for [tex]\( m \)[/tex]:
[tex]\[ m = \frac{PE}{g \cdot h} \][/tex]
Let's plug in the numbers:
[tex]\[ m = \frac{235200}{9.81 \cdot 30} \][/tex]
Next, we perform the multiplication in the denominator:
[tex]\[ 9.81 \cdot 30 = 294.3 \][/tex]
Now, we divide the potential energy by this product:
[tex]\[ m = \frac{235200}{294.3} \approx 799.18 \, \text{kg} \][/tex]
So, the mass of the roller coaster is approximately [tex]\( 799.18 \, \text{kg} \)[/tex], which is closest to the option [tex]\( 800 \, \text{kg} \)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{800 \, \text{kg}}
\][/tex]
We are given the potential energy, height, and the acceleration due to gravity. We need to find the mass of the roller coaster.
The formula for potential energy (PE) is:
[tex]\[ PE = m \cdot g \cdot h \][/tex]
Where:
- [tex]\( PE \)[/tex] is the potential energy (in joules),
- [tex]\( m \)[/tex] is the mass (in kilograms) that we need to find,
- [tex]\( g \)[/tex] is the acceleration due to gravity (approximately [tex]\( 9.81 \, \text{m/s}^2 \)[/tex]),
- [tex]\( h \)[/tex] is the height (in meters).
Given:
- [tex]\( PE = 235200 \, \text{J} \)[/tex]
- [tex]\( h = 30 \, \text{m} \)[/tex]
- [tex]\( g = 9.81 \, \text{m/s}^2 \)[/tex]
We need to rearrange the formula to solve for [tex]\( m \)[/tex]:
[tex]\[ m = \frac{PE}{g \cdot h} \][/tex]
Let's plug in the numbers:
[tex]\[ m = \frac{235200}{9.81 \cdot 30} \][/tex]
Next, we perform the multiplication in the denominator:
[tex]\[ 9.81 \cdot 30 = 294.3 \][/tex]
Now, we divide the potential energy by this product:
[tex]\[ m = \frac{235200}{294.3} \approx 799.18 \, \text{kg} \][/tex]
So, the mass of the roller coaster is approximately [tex]\( 799.18 \, \text{kg} \)[/tex], which is closest to the option [tex]\( 800 \, \text{kg} \)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{800 \, \text{kg}}
\][/tex]
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