Let's solve each question step by step.
### Question 5:
What is the mass of a rock lifted 2 meters off the ground that has [tex]$196 J$[/tex] of potential energy?
The formula for gravitational potential energy (PE) is:
[tex]\[ \text{PE} = m \cdot g \cdot h \][/tex]
where:
- [tex]\( \text{PE} \)[/tex] is the potential energy,
- [tex]\( m \)[/tex] is the mass of the object,
- [tex]\( g \)[/tex] is the acceleration due to gravity (approximately [tex]\[9.8 \, m/s^2\][/tex]),
- [tex]\( h \)[/tex] is the height.
We are given:
[tex]\[ \text{PE} = 196 \, J \][/tex]
[tex]\[ h = 2 \, m \][/tex]
[tex]\[ g = 9.8 \, m/s^2 \][/tex]
Rearranging the formula to solve for [tex]\( m \)[/tex] gives:
[tex]\[ m = \frac{\text{PE}}{g \cdot h} \][/tex]
Substitute the given values:
[tex]\[ m = \frac{196 \, J}{9.8 \, m/s^2 \cdot 2 \, m} \][/tex]
[tex]\[ m = \frac{196}{19.6} \][/tex]
[tex]\[ m = 10 \, kg \][/tex]
So, the mass of the rock is [tex]\( 10 \, kg \)[/tex].
The correct answer is C. [tex]$10 kg$[/tex].
### Question 6:
A [tex]$180 kg$[/tex] roller coaster car is at the peak of its track, at 60 meters tall. How much potential energy does the car hold?
We again use the formula for gravitational potential energy:
[tex]\[ \text{PE} = m \cdot g \cdot h \][/tex]
We are given:
[tex]\[ m = 180 \, kg \][/tex]
[tex]\[ h = 60 \, m \][/tex]
[tex]\[ g = 9.8 \, m/s^2 \][/tex]
Substitute the given values into the formula:
[tex]\[ \text{PE} = 180 \, kg \cdot 9.8 \, m/s^2 \cdot 60 \, m \][/tex]
[tex]\[ \text{PE} = 180 \cdot 9.8 \cdot 60 \][/tex]
[tex]\[ \text{PE} = 105840 \, J \][/tex]
So, the potential energy of the car is [tex]\( 105840 \, J \)[/tex].
The correct answer is C. [tex]$1.06 \times 10^5 J$[/tex].
