I have a diagram of a roller coaster. It starts at a height of 100m where the v=0, it then drops completely to the ground and then goes into a loop that is 75m high; the point where the loop starts is point A. From the loop, the track raises to e height of 50m (this is point B) and then the track goes up to 80m, there, another loop forms (this is point C) and then the track goes up again to now a height of 90m at point D.

Question

I have a diagram of a roller coaster. It starts at a height of 100m where the v=0, it then drops completely to the ground and then goes into a loop that is 75m high; the point where the loop starts is point A. From the loop, the track raises to e height of 50m (this is point B) and then the track goes up to 80m, there, another loop forms (this is point C) and then the track goes up again to now a height of 90m at point D.

Find the velocity of the roller coaster at each point.
I'm not sure how to do this because I don't have the mass to do GPE=mgh. Please help.

Answer 1

You don't need the mass. Write the conservation of energy equations and you will see than the mass m cancesl out.

Answer 2

I tried doing that; can you please explain.

Wait, so, would the first velocity be mgh=1/2mv^2 and then I take away the m's?

Answer 3

Also, I think I've found the first velocity, but would I disregard that when finding the velocity at point B?

Answer 4

yes.

The sum of V^2/2 (KE/m) and gh (GPE/m) is a constant everywhere.
(This method neglects friction and the rotational kinetic energy of the wheels, two assumptions they probably expect you to make.)

Answer 5

oh okay, thank you very much. :)

Answer 6

To find the velocity of the roller coaster at each point, we can use the principle of conservation of mechanical energy. This principle states that the sum of the kinetic energy (KE) and potential energy (PE) at any point remains constant, assuming there are no non-conservative forces (such as friction or air resistance) acting on the roller coaster.

Let's analyze each point individually:

1. Point A:
At point A, the roller coaster starts at a height of 100m with zero velocity. Therefore, the initial potential energy (PE_initial) is given by PE_initial = m * g * h_initial, where m is the mass of the roller coaster, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h_initial is the initial height. As you mentioned, we do not have the mass of the roller coaster, but for this analysis, we can assume it cancels out since we are comparing different points on the roller coaster.

Due to conservation of mechanical energy, the initial potential energy is equal to the final kinetic energy (KE_final) at this point, which is given by KE_final = (1/2) * m * v^2, where v is the velocity of the roller coaster at point A.

Setting these equations equal to each other and solving for v, we get:

m * g * h_initial = (1/2) * m * v^2
v^2 = 2 * g * h_initial
v = sqrt(2 * g * h_initial)

Substituting the values, v = sqrt(2 * 9.8 * 100) = 44.3 m/s (approximately)

2. Point B:
At point B, the roller coaster is at a height of 50m. Using the same equation as above:

v^2 = 2 * g * h_b
v = sqrt(2 * 9.8 * 50) = 31.3 m/s (approximately)

3. Point C:
At point C, the roller coaster is at a height of 80m. Using the same equation:

v^2 = 2 * g * h_c
v = sqrt(2 * 9.8 * 80) = 39.6 m/s (approximately)

4. Point D:
At point D, the roller coaster is at a height of 90m. Using the same equation:

v^2 = 2 * g * h_d
v = sqrt(2 * 9.8 * 90) = 42.0 m/s (approximately)

Please note that these calculations assume ideal conditions without accounting for factors such as friction or air resistance.