To determine the mass of a ball using its gravitational potential energy, we can use the formula for gravitational potential energy, which is given by:
[tex]\[ PE = m \cdot g \cdot h \][/tex]
Where:
- [tex]\( PE \)[/tex] is the gravitational potential energy, in Joules (J).
- [tex]\( m \)[/tex] is the mass, in kilograms (kg).
- [tex]\( g \)[/tex] is the acceleration due to gravity, approximately [tex]\( 9.81 \, m/s^2 \)[/tex].
- [tex]\( h \)[/tex] is the height above the ground, in meters (m).
Given values:
- [tex]\( PE = 116.62 \, J \)[/tex]
- [tex]\( h = 85 \, m \)[/tex]
- [tex]\( g = 9.81 \, m/s^2 \)[/tex]
We need to solve for the mass ([tex]\( m \)[/tex]). Rearrange the formula to solve for [tex]\( m \)[/tex]:
[tex]\[ m = \frac{PE}{g \cdot h} \][/tex]
Substitute the given values into this equation:
[tex]\[ m = \frac{116.62 \, J}{9.81 \, m/s^2 \cdot 85 \, m} \][/tex]
Perform the calculation:
[tex]\[ m \approx 0.1398572884811417 \, kg \][/tex]
Rounding to two decimal places, the mass [tex]\( m \)[/tex] is approximately [tex]\( 0.14 \, kg \)[/tex].
So, the mass of the ball is:
[tex]\[ 0.14 \, kg \][/tex]
This matches the first option:
[tex]\[ \boxed{0.14 \, kg} \][/tex]
