A 540 kg roller coaster starts from rest at point A (Fig. 6.29) and rolls freely (no friction) to point B where the brakes are applied and it slides along horizontally with a frictional force of 520 N. How far does the coaster slide past point B before coming to rest? Height of A is 30m height of B is 10m
m * g * h = f * d
540 * 9.8 * 20 = 520 * d
To solve this problem, we need to calculate the initial potential energy of the roller coaster at point A, then find the velocity at point B using the principle of conservation of energy, and finally use the frictional force to calculate the distance the roller coaster slides past point B.
Step 1: Calculate the potential energy at point A.
The potential energy at point A can be calculated using the formula:
Potential Energy = mass x gravity x height
Given:
mass (m) = 540 kg
gravity (g) = 9.8 m/s^2
height at A (hA) = 30 m
Potential Energy at A = 540 kg x 9.8 m/s^2 x 30 m
= 158,760 J
Step 2: Find the velocity at point B.
According to the principle of conservation of energy, the initial potential energy at point A will be converted into kinetic energy at point B.
The kinetic energy at point B can be calculated using the formula:
Kinetic Energy (KB) = (1/2) x mass x velocity^2
The kinetic energy at B is equal to the initial potential energy at A, so we can set up the equation:
Potential Energy at A = Kinetic Energy at B
158,760 J = (1/2) x mass x velocity^2
Solving for the velocity at B:
velocity^2 = (2 x Potential Energy at A) / mass
velocity^2 = (2 x 158,760 J) / 540 kg
velocity^2 = 586.67 m^2/s^2
velocity = √586.67 m/s
velocity ≈ 24.2 m/s
Step 3: Calculate the distance the coaster slides past point B.
At point B, the only force acting on the roller coaster is friction, so the frictional force can be used to calculate the acceleration.
Using Newton's second law, we can calculate the acceleration at B:
Frictional Force = mass x acceleration
520 N = 540 kg x acceleration
acceleration = 520 N / 540 kg
acceleration ≈ 0.963 m/s^2
To find the distance traveled, we can use the equation of motion:
v^2 = u^2 + 2a(s - u)
where:
v = final velocity (0 m/s)
u = initial velocity (24.2 m/s)
a = acceleration (-0.963 m/s^2)
s = distance
Rearranging the equation, we have:
s = (v^2 - u^2) / (2a)
s = (0^2 - (24.2 m/s)^2) / (2 x (-0.963 m/s^2))
s = (-586.67 m^2/s^2) / (-1.926 m/s^2)
s ≈ 304 m
Therefore, the roller coaster slides approximately 304 meters past point B before coming to rest.
To calculate the distance that the roller coaster slides past point B before coming to rest, we need to use the conservation of mechanical energy. Here's how you can solve this problem:
Step 1: Calculate the potential energy at point A.
The potential energy at point A can be calculated using the formula:
Potential Energy = Mass × Gravitational Acceleration × Height
Given that the mass of the roller coaster is 540 kg and the height of point A is 30 m, the potential energy at point A is:
Potential Energy at A = 540 kg × 9.8 m/s^2 × 30 m
Step 2: Calculate the potential energy at point B.
The potential energy at point B can also be calculated using the formula mentioned earlier:
Potential Energy at B = Mass × Gravitational Acceleration × Height
Given that the height of point B is 10 m, the potential energy at point B is:
Potential Energy at B = 540 kg × 9.8 m/s^2 × 10 m
Step 3: Find the difference in potential energy between A and B.
The difference in potential energy can be calculated by subtracting the potential energy at B from the potential energy at A:
ΔPotential Energy = Potential Energy at A - Potential Energy at B
Step 4: Convert the potential energy to kinetic energy.
Since the roller coaster starts from rest at point A, the potential energy difference calculated in the previous step will be converted entirely into kinetic energy. Therefore, ΔPotential Energy = ΔKinetic Energy.
Step 5: Calculate the velocity at point B.
The kinetic energy at point B can be calculated using the formula:
Kinetic Energy = (1/2) × Mass × Velocity^2
Given that the mass of the roller coaster is 540 kg and the velocity at point B is unknown, we can rewrite the equation as:
ΔKinetic Energy = (1/2) × 540 kg × Velocity^2
Step 6: Set up the equation for work done by friction.
The work done by friction can be calculated using the formula:
Work = Force × Distance
Given that the force of friction is 520 N and the distance we want to find is D, we can write the equation as:
Work = 520 N × D
Step 7: Use the Work-Energy theorem.
According to the Work-Energy theorem, the work done on an object is equal to the change in its kinetic energy. Therefore, we can set up the equation:
520 N × D = (1/2) × 540 kg × Velocity^2
Step 8: Solve for Velocity.
Rearrange the equation from the previous step to solve for the velocity at point B:
Velocity^2 = (2 × 520 N × D) / (540 kg)
Velocity = √((2 × 520 N × D) / (540 kg))
Step 9: Use the equation for the final velocity and solve for D.
Since the roller coaster comes to rest at point B, the final velocity is zero. Therefore, we can set up the equation as:
0 = √((2 × 520 N × D) / (540 kg))
Step 10: Solve for D.
Square both sides of the equation and solve for D:
0 = (2 × 520 N × D) / (540 kg)
0 = (1040 N × D) / (540 kg)
1040 N × D = 0
D = 0
Therefore, the coaster slides zero distance past point B before coming to rest.