Energy conservation with conservative forces: A car on a roller coaster starts at zero speed at an elevation above the ground of 26 m. It coasts down a slope, and then climbs a hill. The top of the hill is at an elevation of 16 m. What is the speed of the car at the top of the hill? Neglect any frictional effects.

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Quizplus · Accepted Answer

The conservation of mechanical energy principle states that the total mechanical energy in a system (kinetic plus potential energy) remains constant if only conservative forces are acting. Since there's no friction, we can apply this principle here. The car's initial potential energy at 26 m is converted into kinetic energy and potential energy at 16 m. Using the formula for potential energy (PE = mgh) and kinetic energy (KE = 0.5mv^2), and setting the initial total energy equal to the final total energy, we can solve for the car's speed at the top of the hill.Initial energy (all potential) = mgh₁, where h₁ = 26 m.Final energy = mgh₂ + 0.5mv^2, where h₂ = 16 m.Setting initial energy equal to final energy and solving for v gives us:mgh₁ = mgh₂ + 0.5mv^2g(h₁ - h₂) = 0.5v^2v^2 = 2g(h₁ - h₂)v = sqrt(2g(h₁ - h₂))Substituting g = 9.8 m/s^2, h₁ = 26 m, and h₂ = 16 m:v = sqrt(2*9.8*(26-16))v = sqrt(2*9.8*10)v = sqrt(196)v = 14 m/s

Energy Conservation with Conservative Forces: a Car on a Roller