The minimum velocity required at the bottom of the loop is closest to 31.4 m/s. Hence the correct option is c.
At the top of the loop, the normal force acting on the coaster car is zero. To prevent the coaster from falling at this point, the vertical component of the car's velocity must be enough to overcome gravity.
We can analyze this problem using the concept of mechanical energy. At the bottom of the loop, the coaster car has a certain amount of kinetic energy due to its speed. As it travels up the loop, this kinetic energy is converted into gravitational potential energy. At the top of the loop, all the kinetic energy is converted to potential energy, because the car momentarily comes to a stop (relative to its vertical motion) due to zero normal force.
Let's denote the minimum velocity required at the bottom of the loop as v. At the bottom of the loop, the coaster car is at a height of 0 meters relative to the bottom of the loop. At the top of the loop, the coaster car is at a height of 2R relative to the bottom of the loop.
Using the principle of conservation of mechanical energy, we can equate the kinetic energy at the bottom of the loop to the gravitational potential energy at the top of the loop.
Kinetic Energy (KE) = 1/2 * m * v^2
Gravitational Potential Energy (GPE) = m * g * h
where:
m is the mass of the coaster car (which cancels out because it appears in both terms)
v is the velocity at the bottom of the loop
g is the acceleration due to gravity
h is the height relative to the bottom of the loop
Equating these:
1/2 * m * v^2 = m * g * 2R
v^2 = 4 * g * R
v = sqrt(4 * g * R)
Plugging in the values:
v = sqrt(4 * 9.81 m/s^2 * 15 m)
v = approximately 31.4 m/s
Therefore, the minimum velocity required at the bottom of the loop is closest to 31.4 m/s. Hence the correct option is c.