To determine the minimum speed the roller coaster can have without falling from the track, we can use the concept of centripetal force.
Centripetal force is the force that keeps an object moving in a circular path. It is given by the formula:
F_centripetal = (m * v^2) / r
Where:
F_centripetal is the centripetal force,
m is the mass of the roller coaster,
v is the velocity of the roller coaster, and
r is the radius of curvature of the track.
In this case, we want to find the minimum speed, so we consider the roller coaster at the point where it is just about to fall off the track. At this point, the centripetal force is equal to the force of gravity.
Let's assume the mass of the roller coaster is m.
The force of gravity is given by the formula:
F_gravity = m * g
Where:
g is the acceleration due to gravity.
Since the centripetal force and the force of gravity are equal at the point of almost falling off the track, we can set the two equations equal to each other:
(m * v^2) / r = m * g
Simplifying the equation, we have:
v^2 = r * g
Now we can solve for the minimum speed (v) by taking the square root of both sides of the equation:
v = sqrt(r * g)
Given that the radius of curvature of the track is 32.1 m and the acceleration due to gravity is approximately 9.8 m/s^2, we can plug in these values into the equation to find the minimum speed:
v = sqrt(32.1 * 9.8)
Calculating the value, we find:
v ≈ 17.7 m/s
Therefore, the minimum speed the roller coaster can have without falling from the track is approximately 17.7 m/s.