The loaded coaster has a mass of 2000
kg. The starting point of the run (B) is 25 m above point A; the run from B to C is 120 m long,
the run from C to D is 150 m. At D, brakes are applied, bringing the coaster to a halt X m further
along, at E. The average frictional force between B and D is 500 N. Find (a) the work required
to raise the cart to its starting point B (assume velocity at B is ~ zero); (b) the speed of the roller
coaster at D; (c) if the average braking force applied between D and E is 50 kN, what is X?
To solve this problem, we need to use principles of work, energy, and Newton's second law. Let's break down each part of the question and find the answers step by step.
(a) To find the work required to raise the cart to its starting point B, we can use the formula for gravitational potential energy:
Work = Mass * Gravity * Height
Given the mass of the coaster (2000 kg) and the height from A to B (25 m), we can calculate the work required.
Work = 2000 kg * 9.8 m/s^2 * 25 m
Work = 490,000 J (Joules)
Therefore, the work required to raise the cart to its starting point B is 490,000 Joules.
(b) To find the speed of the roller coaster at point D, we need to solve for the kinetic energy at that point. The total mechanical energy at point D is equal to the sum of the gravitational potential energy at point B and the work done by friction (negative).
Total mechanical energy at D = Gravitational potential energy at B - Work done by friction
Gravitational potential energy at B = Mass * Gravity * Height
Gravitational potential energy at B = 2000 kg * 9.8 m/s^2 * 25 m
Work done by friction = Frictional force * Distance
Work done by friction = 500 N * (120 m + 150 m)
Total mechanical energy at D = 2000 kg * 9.8 m/s^2 * 25 m - 500 N * (120 m + 150 m)
Now, we can equate the total mechanical energy to the kinetic energy at point D:
Total mechanical energy at D = (1/2) * Mass * Velocity^2
From here, solve for velocity:
Velocity = √((2 * Total mechanical energy at D) / Mass)
Substitute the values into the equation and solve for velocity.
(c) To find the distance X, we need to use the formula for work done by a force:
Work = Force * Distance
Given the average braking force (50 kN) and the distance from D to E (X), we can calculate the work done.
Work = 50 kN * X
Since the work done must equal the change in kinetic energy between D and E (which is zero because the roller coaster comes to a halt), we can set up the equation:
Work = Change in kinetic energy
Work = 0 J
Therefore:
50 kN * X = 0
Solve for X:
X = 0 / 50 kN
Thus, X is equal to zero.
To recap:
(a) The work required to raise the cart to its starting point B is 490,000 Joules.
(b) The speed of the roller coaster at point D can be calculated by solving the equations mentioned above.
(c) The distance X is equal to zero, since there is no additional distance covered after point D where the roller coaster comes to a halt.