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

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

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


Sagot :

To find the mass of the roller coaster, we need to use the formula for potential energy, which is given by:

[tex]\[ PE = m \cdot g \cdot h \][/tex]

Where:
- [tex]\( PE \)[/tex] is the potential energy,
- [tex]\( m \)[/tex] is the mass,
- [tex]\( g \)[/tex] is the acceleration due to gravity, and
- [tex]\( h \)[/tex] is the height.

We are given the following values:
- [tex]\( PE = 235,200 \)[/tex] joules,
- [tex]\( g = 9.81 \)[/tex] m/s²,
- [tex]\( h = 30 \)[/tex] meters.

We need to solve for the mass [tex]\( m \)[/tex]. We can rearrange the formula to isolate [tex]\( m \)[/tex]:

[tex]\[ m = \frac{PE}{g \cdot h} \][/tex]

Substituting in the given values:

[tex]\[ m = \frac{235,200}{9.81 \cdot 30} \][/tex]

Calculating the denominator first:

[tex]\[ 9.81 \cdot 30 = 294.3 \][/tex]

Now, divide the potential energy by the product of [tex]\( g \)[/tex] and [tex]\( h \)[/tex]:

[tex]\[ m = \frac{235,200}{294.3} \][/tex]

Performing the division:

[tex]\[ m \approx 799.18 \, \text{kg} \][/tex]

Therefore, the mass of the roller coaster is approximately [tex]\( 799.18 \)[/tex] kg. Among the given answer choices, the closest value is:

[tex]\[ 800 \, \text{kg} \][/tex]

So, the mass of the roller coaster is closest to [tex]\( 800 \)[/tex] kg.
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