A 775 kg roller coaster car is at the top of a roller coaster car is at the top of a roller coaster as shown below. The height of the car at point A is the minimum height that is necessary for the car to stay on the loop. The radius of the loop is 30 m.
What is the height of the car at point A?
g = v^2/r at top of loop
g = v^2/30
solve for v^2 = 30 *g
now the car drops A-2r to that point at the top of the loop
(1/2) m v^2 = m g (A-60)
v^2 = 2 g(A-60)
30 g = 2g (A-60 ) = 2 g A -120 g
150 g = 2gA
A = 75
check my arithmetic !
Note this does no depend on the mass (lucky or it would be different for skinny kids and fat adults), or even on g (independent of planet:)
check, drops 15 meters to top of loop
v^2 = 2 g (15)
needs v^2/30 = g
sure enough 2*15 g /30 = g
To determine the height of the car at point A, we need to use the concept of conservation of energy.
1. Identify the potential energy at the top of the loop (point A):
The potential energy at the top of the loop can be calculated using the formula: P.E = m * g * h, where m is the mass of the car (775 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the car at point A.
2. Determine the kinetic energy required to complete the loop:
To stay on the loop, the car needs a minimum of zero net normal force at the top of the loop. This means that the centripetal force is equal to the force of gravity. The centripetal force is given by the formula: Fc = m * v^2 / r, where m is the mass of the car (775 kg), v is the velocity of the car, and r is the radius of the loop (30 m).
By setting the centripetal force equal to the force of gravity, we get: m * v^2 / r = m * g.
Rearranging the equation gives: v^2 = r * g.
We can then calculate the velocity required using the equation: v = √(r * g).
3. Calculate the kinetic energy:
The kinetic energy of the car is given by the formula: K.E = (1/2) * m * v^2.
4. Equate the potential energy to the kinetic energy:
P.E = K.E.
5. Solve for the height of the car at point A (h):
Substitute the values of potential energy (P.E) and kinetic energy (K.E) into the equation P.E = K.E, and solve for h.
Using these steps, we can determine the height of the car at point A.