This problem is similar to the previous.
Again, use conservation of energies.
Again, use conservation of energies.
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.