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?
A.The car's maximum gravitational potential energy occurs at the start, at x = 0
B.At the end, at x = 200, the car has only kinetic energy
C.Mechanical energy of the roller-coaster is not conserved because of the presence of the normal force
D.At the end, at x = 200, the car has the same total energy that it began with
E.In the middle of the dip, at x = 100, the car has both gravitational potential energy and kinetic energy
A. False. The car's maximum gravitational potential energy occurs at the top of the roller-coaster, when it is at its highest point.
B. False. The car will have both kinetic energy and gravitational potential energy at the end of the roller-coaster.
C. False. The mechanical energy of the roller-coaster is conserved, as there is no external force acting on it.
D. True. The total mechanical energy of the car is conserved throughout the roller-coaster ride, so it will have the same total energy at the end as it had at the beginning.
E. True. In the middle of the dip, at x = 100, the car will have both gravitational potential energy and kinetic energy.
To analyze the given statements, let's look at each one individually:
A. The car's maximum gravitational potential energy occurs at the start, at x = 0.
This statement is true. At the start, the car is at its highest point, so it has the maximum gravitational potential energy.
B. At the end, at x = 200, the car has only kinetic energy.
This statement is false. At the end, at x = 200, the car will have both kinetic energy and gravitational potential energy (although less than at the start).
C. Mechanical energy of the roller-coaster is not conserved because of the presence of the normal force.
This statement is false. The mechanical energy of the roller-coaster is conserved because it is a frictionless roller-coaster. The normal force does not affect the conservation of mechanical energy.
D. At the end, at x = 200, the car has the same total energy that it began with.
This statement is false. At the end, at x = 200, the car will have less total energy than it began with because some of its initial gravitational potential energy is converted into other forms of energy, such as kinetic energy.
E. In the middle of the dip, at x = 100, the car has both gravitational potential energy and kinetic energy.
This statement is true. In the middle of the dip, at x = 100, the car will have both gravitational potential energy (although less than at the start) and kinetic energy.
In summary, the true statements are:
A. The car's maximum gravitational potential energy occurs at the start, at x = 0.
E. In the middle of the dip, at x = 100, the car has both gravitational potential energy and kinetic energy.
To answer these questions, let's analyze the different points along the roller coaster and discuss the concept of mechanical energy.
At x = 0, the car is at the highest point with a height of y = H. Here, the car has gravitational potential energy due to its elevation above the reference level (y = 0). So, statement A is incorrect. The car has maximum potential energy at the start.
As the car goes down into the dip and reaches x = 100 with a height of y = 0, it has lost all of its initial potential energy. But as it descends, it gains kinetic energy due to its increasing speed. Therefore, at x = 100, the car has kinetic energy and no potential energy. Thus, statement E is correct.
Now, let's consider statement B. At x = 200, where the car reaches a height of y = H/2, it has gained potential energy back but lost some kinetic energy. Thus, it has both potential and kinetic energy at this point. Therefore, statement B is incorrect.
Next, let's discuss statement C. The mechanical energy of an object refers to the sum of its kinetic energy and potential energy. In the absence of non-conservative forces like friction, the mechanical energy of the object is conserved. The normal force is a conservative force, meaning it does not dissipate energy. Therefore, the mechanical energy of the roller-coaster is conserved, and statement C is incorrect.
Finally, let's address statement D. Since the mechanical energy is conserved (as discussed above), the car will have the same total energy at the end, x = 200, as it did at the start, x = 0. So, statement D is correct.
In summary:
- Statement A is incorrect.
- Statement B is incorrect.
- Statement C is incorrect.
- Statement D is correct.
- Statement E is correct.