A roller coaster with a potential energy of [tex]$235,200 \, J[tex]$[/tex] sits at the top of a [tex]$[/tex]30 \, m$[/tex] high hill. What is the mass of the coaster?

(Formula: [tex]PE = mgh[/tex])

A. [tex]800 \, kg[/tex]
B. [tex]7,840 \, kg[/tex]
C. [tex]8,000 \, kg[/tex]
D. [tex]78,400 \, kg[/tex]



Answer :

Certainly! Let's solve this problem step by step using the given formula for potential energy:

The formula for potential energy (PE) is:
[tex]\[ PE = m \cdot g \cdot h \][/tex]

Where:
- \( PE \) is the potential energy,
- \( m \) is the mass,
- \( g \) is the gravitational acceleration, which is \( 9.8 \, \text{m/s}^2 \),
- \( h \) is the height.

Given the values:
[tex]\[ PE = 235,200 \, \text{J} \][/tex]
[tex]\[ h = 30 \, \text{m} \][/tex]
[tex]\[ g = 9.8 \, \text{m/s}^2 \][/tex]

We need to find the mass \( m \). To do this, we rearrange the formula to solve for \( m \):
[tex]\[ m = \frac{PE}{g \cdot h} \][/tex]

Now, substitute the known values into the equation:
[tex]\[ m = \frac{235,200 \, \text{J}}{9.8 \, \text{m/s}^2 \cdot 30 \, \text{m}} \][/tex]

Calculate the denominator first:
[tex]\[ 9.8 \, \text{m/s}^2 \cdot 30 \, \text{m} = 294 \, \text{m}^2/\text{s}^2 \][/tex]

Now divide the potential energy by this product:
[tex]\[ m = \frac{235,200 \, \text{J}}{294 \, \text{m}^2/\text{s}^2} \][/tex]

Finally, compute the division:
[tex]\[ m = 800 \, \text{kg} \][/tex]

So, the mass of the coaster is:
[tex]\[ 800 \, \text{kg} \][/tex]

Thus, the correct answer is:
[tex]\[ 800 \, \text{kg} \][/tex]