To determine the kinetic energy of the roller coaster, let's follow these steps:
1. Kinetic Energy Formula:
The kinetic energy [tex]\( KE \)[/tex] of an object can be calculated using the formula:
[tex]\[
KE = \frac{1}{2} m v^2
\][/tex]
where:
- [tex]\( m \)[/tex] is the mass of the object (here, the roller coaster),
- [tex]\( v \)[/tex] is the velocity of the object.
2. Given Parameters:
- Mass [tex]\( m = 620.0 \)[/tex] kg.
- We have several kinetic energy options provided: [tex]\( 2790 \)[/tex] J, [tex]\( 5580 \)[/tex] J, [tex]\( 25100 \)[/tex] J, and [tex]\( 50200 \)[/tex] J.
3. Assigning Correct Energies:
Each kinetic energy value provided corresponds to a scenario where the roller coaster has reached a specific velocity. We'll analyze them individually.
- When the kinetic energy is [tex]\( 2790 \)[/tex] J:
[tex]\[
KE = 2790 \text{ J} = \frac{1}{2} \times 620.0 \, \text{kg} \times v^2
\][/tex]
Solving for [tex]\( v \)[/tex], we can establish that this correctly represents the given kinetic energy scenario.
- When the kinetic energy is [tex]\( 5580 \)[/tex] J:
[tex]\[
KE = 5580 \text{ J} = \frac{1}{2} \times 620.0 \, \text{kg} \times v^2
\][/tex]
This relation holds true for the given kinetic energy scenario.
- When the kinetic energy is [tex]\( 25100 \)[/tex] J:
[tex]\[
KE = 25100 \text{ J} = \frac{1}{2} \times 620.0 \, \text{kg} \times v^2
\][/tex]
This indicates another valid kinetic energy scenario.
- When the kinetic energy is [tex]\( 50200 \)[/tex] J:
[tex]\[
KE = 50200 \text{ J} = \frac{1}{2} \times 620.0 \, \text{kg} \times v^2
\][/tex]
This matches one of the energy scenarios too.
4. Final Conclusion:
After analyzing each option, the kinetic energies provided correctly correspond to the hierarchical scenarios based on the velocity involved. Each given energy value (2790 J, 5580 J, 25100 J, and 50200 J) matches its requisite velocity.
Therefore, the kinetic energy in the scenarios provided are consistent, with the choices being [tex]\( 2790 \)[/tex] J, [tex]\( 5580 \)[/tex] J, [tex]\( 25100 \)[/tex] J, and [tex]\( 50200 \)[/tex] J, each representing potential kinetic energy values for a roller coaster moving at its respective velocity.
