how fast must a roller coaster be going to get around a vertical loop with a radius of 15 m
To determine how fast a roller coaster must be going to get around a vertical loop with a radius of 15 m, we can use the concept of centripetal force.
The centripetal force required for an object to move in a circular path is given by:
Fc = (m * v^2) / r
Where Fc is the centripetal force, m is the mass of the roller coaster, v is the velocity, and r is the radius of the loop.
In this case, the centripetal force required to keep the roller coaster moving in a vertical loop is equal to the gravitational force acting on the roller coaster:
Fc = m * g
Where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Setting these two equations equal to each other and solving for v, we have:
(m * v^2) / r = m * g
Canceling out the mass, we get:
v^2 = r * g
Taking the square root of both sides, we can solve for v:
v = √(r * g)
Plugging in the given values, we have:
v = √(15 m * 9.8 m/s^2)
v = √(147 m^2/s^2)
v = 12.124 m/s
Therefore, the roller coaster must be traveling at a speed of approximately 12.124 m/s (or about 43.65 km/h) to successfully make it around a vertical loop with a radius of 15 m.