Question

fernando is an engineer at an amusement park who is experimenting with changes to the setup for a magnetic roller coaster ride. in one ride, there are two identical roller coaster cars (red and yellow) that start on opposite sides of a large magnet located in the center of a station. he moves both cars, so each car ends up one space farther away from the large magnet. how did the potential energy of the two cars change? did the potential energy of one car change more than the other? why do you think so?

Explanation:

Step1: Analyze the nature of magnetic potential energy

Magnetic potential energy depends on the distance from the magnet and the properties of the object (in this case, the cars are identical). When moving away from a magnet, the magnetic potential energy changes.

Step2: Consider the identical nature of the cars

Since the two roller - coaster cars are identical (same mass, same magnetic properties etc.) and they are moved the same distance (one space) from the magnet.

Step3: Apply the formula for magnetic potential energy (qualitative)

The formula for magnetic potential energy \(U = - \vec{m}\cdot\vec{B}\) (where \(\vec{m}\) is the magnetic moment and \(\vec{B}\) is the magnetic field). For objects moving in a non - uniform magnetic field (assuming the field from the large magnet is non - uniform), the change in potential energy \(\Delta U\) is related to the work done against the magnetic force. Since \(F =
abla(\vec{m}\cdot\vec{B})\) and \(W=\int F\cdot dx\) (for a one - dimensional displacement \(dx\) away from the magnet). For identical cars (\(\vec{m}\) is the same) and same \(dx\) (one space), the work done (and thus the change in potential energy) is the same.

Answer:

The potential energy of both cars increased. The potential energy of neither car changed more than the other. Because the two cars are identical (have the same magnetic properties, mass etc.) and they are moved the same distance away from the magnet. So, the change in their magnetic potential energies is the same.