# A 500kg roller coaster starts from rest at points 35m above the ground level. The car drives down into a valley 4m above the ground and then climbs the top of a hill that is 24m above the ground level. What is the velocity of the car in the valley and at the top of the hill? (Neglect friction)

## This problem is similar to the previous.

Again, use conservation of energies.

## To determine the velocity of the roller coaster in the valley and at the top of the hill, we can use the principle of conservation of mechanical energy. According to this principle, the total mechanical energy of a system remains constant if no external forces (like friction) are acting on it. The total mechanical energy is the sum of the potential energy (PE) and the kinetic energy (KE) of the roller coaster.

First, let's analyze the roller coaster's motion at the valley. At this point, the roller coaster has a potential energy (PE) of 4m above the ground level. As it descends, this potential energy is converted into kinetic energy (KE). At the bottom of the valley, all the potential energy is converted into kinetic energy, and the roller coaster has no more potential energy. Therefore, the kinetic energy of the roller coaster at the valley is equal to the initial potential energy:

KE_valley = PE_initial

Next, let's calculate the potential energy at the top of the hill. Here, the roller coaster has a potential energy (PE) of 24m above the ground level. As it climbs up, this potential energy is converted into kinetic energy (KE). At the top of the hill, all the potential energy is converted into kinetic energy, and the roller coaster has no more potential energy. Therefore, the kinetic energy of the roller coaster at the top of the hill is equal to the potential energy it had initially plus the potential energy at the top of the hill:

KE_hill = PE_initial + PE_hill

Since we know the mass of the roller coaster (m = 500kg) and the acceleration due to gravity (g = 9.8m/s^2), we can use the following formulas to find the velocities:

KE = (1/2)mv^2 (kinetic energy formula)
PE = mgh (potential energy formula)

To find the velocity (v), we can rearrange the kinetic energy formula:

v = sqrt((2KE) / m)

Now, let's calculate the velocities at the valley and the top of the hill:

1. Velocity at the valley:

Substituting KE_valley = PE_initial into the velocity formula:

v_valley = sqrt((2PE_initial) / m)

2. Velocity at the top of the hill:

Substituting KE_hill = PE_initial + PE_hill into the velocity formula:

v_hill = sqrt((2(PE_initial + PE_hill)) / m)

By plugging in the values for PE_initial (potential energy at 35m), PE_valley (potential energy at 4m), PE_hill (potential energy at 24m), and m (mass of the roller coaster), we can calculate the velocities at the valley and the top of the hill using the above formulas.