True or false
Consider a simple frictionless roller-coaster. It begins at x = 0, at a height of y = H. Then it drops into a dip, reaching a minimum at x = 100, with zero height (y = 0). Finally, it goes back up at x = 200, but only reaches a height of y = H/2. Assume the roller-coaster car starts from rest at x = 0. Which of the following statements are true?
1)At the end, at x = 200, the car has only gravitational potential energy
2)The car's maximum kinetic energy occurs in the dip, at x = 100
3)Mechanical energy of the roller-coaster is conserved because the normal force does not do work
4)In the middle of the dip, at x = 100, the car has only kinetic energy
5)At the end, at x = 200, the car has more total energy than it began with
1) true if stopped
2) yes max Ke at min height
3) Huh? It is true the normal force does not do work, but the friction force does so mechanical energy is LOST to heat. That is why it ends up lower and has to be towed back up to the start.
4) Well, this is true only if you define zero potential level at the dip, at y = 0
5) of course not.
Let's analyze each statement one by one:
1) At the end, at x = 200, the car has only gravitational potential energy.
False. At any point along the roller-coaster track, the car can have both potential energy and kinetic energy. At x = 200, the car will have both gravitational potential energy and kinetic energy.
2) The car's maximum kinetic energy occurs in the dip, at x = 100.
False. The car's maximum kinetic energy occurs when it reaches the bottom of the dip at x = 100, as it has converted all its potential energy into kinetic energy at that point.
3) Mechanical energy of the roller-coaster is conserved because the normal force does not do work.
True. In the absence of non-conservative forces like friction, mechanical energy (the sum of potential and kinetic energy) is conserved. The normal force does not do any work because it acts perpendicular to the motion of the car, and thus it does not contribute to changing the mechanical energy of the system.
4) In the middle of the dip, at x = 100, the car has only kinetic energy.
False. At the very bottom of the dip, the car has only kinetic energy, as it has converted all its potential energy into kinetic energy. However, in the middle of the dip, the car will have both potential energy and kinetic energy.
5) At the end, at x = 200, the car has more total energy than it began with.
False. In the absence of non-conservative forces, the total energy of the car (the sum of potential and kinetic energy) remains constant throughout the motion. Therefore, at the end (x = 200), the car will have the same total energy as it began with (assuming no energy losses due to other factors).
To determine which of the statements are true, we can analyze the energy transformations that occur throughout the roller-coaster ride.
1) At the end, at x = 200, the car has only gravitational potential energy.
False. At the end of the ride, at x = 200, the car will have both gravitational potential energy and kinetic energy. The potential energy will be lower than at the beginning (x = 0) due to the decrease in height, but the car will still have some kinetic energy.
2) The car's maximum kinetic energy occurs in the dip, at x = 100.
True. In a frictionless situation, the car's maximum kinetic energy will occur when it reaches its lowest point in the dip at x = 100. This is because the potential energy is at a minimum and converted to kinetic energy.
3) Mechanical energy of the roller-coaster is conserved because the normal force does not do work.
True. In a frictionless roller-coaster, the mechanical energy (sum of kinetic energy and potential energy) is conserved. The normal force does not do work because it acts perpendicular to the direction of motion.
4) In the middle of the dip, at x = 100, the car has only kinetic energy.
False. At the midpoint of the dip, the car will have both kinetic energy and gravitational potential energy. The kinetic energy will be at its maximum, but there will still be some potential energy due to the height above the zero level.
5) At the end, at x = 200, the car has more total energy than it began with.
False. In an idealized frictionless system, the total mechanical energy of the roller-coaster remains constant throughout the ride. Therefore, at the end of the ride (x = 200), the car will have the same total energy as at the beginning (x = 0). It may have a different distribution between kinetic and potential energy, but the total energy remains constant.