A roller coaster starts from rest at the top of a steep incline. As it falls, should its velocity depend on how many riders are aboard? Use net work= change in kinetic energy to prove your answer.
To determine if the velocity of a roller coaster depends on the number of riders aboard, we need to analyze the relationship between the net work done on the roller coaster and the change in its kinetic energy.
The work-energy theorem states that the net work done on an object is equal to its change in kinetic energy. Mathematically, it can be written as:
Work = ΔKinetic Energy
In the case of the roller coaster, the net work done on it is the gravitational potential energy (PE) at the top of the incline being converted into kinetic energy (KE) as it falls. Therefore, we can write:
Work = ΔPE = mgh
Here, m represents the mass of the roller coaster, g is the acceleration due to gravity, and h is the height of the incline.
Now, let's consider what would change if the number of riders aboard the roller coaster is varied. The mass of the roller coaster, m, would increase with more riders since their combined weight contributes to the total mass. Therefore, the work done on the roller coaster would increase proportionally.
On the other hand, the change in kinetic energy, ΔKE, is determined solely by the velocity of the roller coaster. The formula for the kinetic energy of an object is given by KE = 0.5 * m * v^2, where v represents the velocity.
Since the mass, m, affects both the work done and the kinetic energy, but cancels out when calculating the ratio of work to change in kinetic energy, we can rewrite the equation as:
Work/m = ΔKE/m
Thus, we can conclude that the number of riders aboard does not affect the velocity of the roller coaster. The mass cancels out when comparing the work done to the change in kinetic energy, indicating that the velocity only depends on the physical characteristics of the roller coaster itself, such as the height of the incline and the force of gravity.