A rollercoaster reaches the top of the steepest hill with a speed of 6.0 km/h. It then descends the hill, which is at an average angle of ?=45 and is 45.0 m long. Estimate its speed when it reaches the bottom. Assume µk=0.18.
To estimate the speed of the rollercoaster when it reaches the bottom of the hill, we can use the principles of conservation of energy and Newton's laws of motion. Here are the steps to get the answer:
1. Determine the gravitational potential energy at the top of the hill:
- The formula for gravitational potential energy is given by U = mgh, where U is the potential energy, m is the mass of the rollercoaster, g is the acceleration due to gravity, and h is the height.
- Since the rollercoaster is at the top of the steepest hill, its height above the bottom of the hill is zero. Therefore, the potential energy at the top is zero.
2. Determine the kinetic energy at the top of the hill:
- The formula for kinetic energy is given by K = (1/2)mv^2, where K is the kinetic energy, m is the mass of the rollercoaster, and v is the velocity.
- We have the velocity at the top, which is 6.0 km/h. However, we need to convert it to meters per second (m/s) to use it in the equation. 1 km/h is equal to 0.2778 m/s, so the velocity is 6.0 x 0.2778 = 1.667 m/s.
- Using the given velocity, we can calculate the kinetic energy at the top.
3. Determine the potential energy and the kinetic energy at the bottom of the hill:
- Since the rollercoaster is at the bottom of the hill, its height above the bottom is 0. Therefore, the potential energy at the bottom is also zero.
- Using the principle of conservation of energy (total energy is conserved), we can say that the total energy at the top (which is just the kinetic energy) is equal to the total energy at the bottom (which is again just the kinetic energy).
- This means that the kinetic energy at the bottom is equal to the kinetic energy at the top.
4. Determine the speed at the bottom of the hill:
- We know that the kinetic energy at the bottom is equal to the kinetic energy at the top.
- By using the formula for kinetic energy, we can solve for the velocity at the bottom.
5. Compensate for friction:
- However, we need to consider the frictional force acting on the rollercoaster due to the coefficient of kinetic friction (µk) given as 0.18.
- Frictional force is given by the equation: Fk = µk * m * g, where Fk is the frictional force, µk is the coefficient of kinetic friction, m is the mass of the rollercoaster, and g is the acceleration due to gravity.
- The work done by friction can be calculated using the equation: Wk = Fk * d * cos(?), where Wk is the work done, Fk is the frictional force, d is the distance traveled which is 45.0 m, and ? is the angle of the hill which is given as 45 degrees.
6. Adjust the velocity at the bottom:
- Subtract the work done by friction (Wk) from the initial kinetic energy to get the adjusted kinetic energy.
- Use this adjusted kinetic energy to find the velocity at the bottom using the kinetic energy formula.
Following these steps will allow us to estimate the speed of the rollercoaster when it reaches the bottom of the hill while accounting for friction.
To estimate the speed of the rollercoaster when it reaches the bottom of the hill, we can use the principles of energy conservation.
Step 1: Convert the initial speed to SI units
The initial speed of the rollercoaster is given as 6.0 km/h. To convert this to m/s, we divide by 3.6 since 1 km/h = 1/3.6 m/s.
So, the initial speed is 6.0 km/h ÷ 3.6 = 1.67 m/s.
Step 2: Determine the change in gravitational potential energy
The rollercoaster descends a hill with an average angle of 45 degrees and a length of 45.0 m.
The change in gravitational potential energy is given by ΔPE = m * g * h, where m is the mass, g is the acceleration due to gravity (9.8 m/s^2), and h is the vertical height.
Since the rollercoaster is on a hill, we need to calculate the vertical height using the average angle and the length of the hill. The vertical height (h) is calculated as h = l * sin(θ), where l is the length of the hill and θ is the angle of the hill.
h = 45.0 m * sin(45 degrees) ≈ 31.8 m
Step 3: Calculate the change in kinetic energy
The change in kinetic energy is given by ΔKE = KE_final - KE_initial.
At the top of the hill, all the initial kinetic energy is converted to potential energy.
So, KE_initial = 0 J.
At the bottom of the hill, all the potential energy is converted back to kinetic energy.
So, ΔPE = - m * g * h = - m * 9.8 m/s^2 * 31.8 m = - 313.56 m * kg / s^2.
Therefore, ΔKE = - ΔPE.
Step 4: Calculate the final kinetic energy
KE_final = ΔKE + KE_initial
= - ΔPE + 0 J
= - (-313.56 m * kg / s^2)
= 313.56 m * kg / s^2
Step 5: Use the work-energy theorem to calculate the final speed
According to the work-energy theorem, the work done on an object is equal to the change in its kinetic energy.
So, W = ΔKE
The work done on the rollercoaster is given by W = F * d, where F is the force and d is the distance over which the force is applied.
The force of friction acting on the rollercoaster is given by F_friction = μk * m * g, where μk is the coefficient of kinetic friction.
Since the work done by the force of friction is negative (it opposes motion), we have:
W = - F_friction * d
Using the equation for work done, we can solve for F_friction:
- F_friction * d = ΔKE
F_friction = - ΔKE / d
= - (313.56 m * kg / s^2) / 45.0 m
= - 6.97 N
The force of friction acting on the rollercoaster is 6.97 N.
The net force acting on the rollercoaster is given by F_net = m * a, where a is the acceleration.
Since the rollercoaster is moving vertically, the net force is the difference between the gravitational force (m * g) and the force of friction (F_friction):
F_net = m * g - F_friction
m * a = m * g - F_friction
a = g - (F_friction / m)
= 9.8 m/s^2 - (6.97 N / m)
≈ 9.8 m/s^2 - (6.97 kg * m / s^2) / m
= 9.8 m/s^2 - 6.97 m/s^2
= 2.83 m/s^2
The acceleration of the rollercoaster is approximately 2.83 m/s^2.
Finally, we can use the equations of motion to calculate the final speed when the rollercoaster reaches the bottom of the hill:
v^2 = u^2 + 2 * a * s
where
v = final speed
u = initial speed (1.67 m/s)
a = acceleration (-2.83 m/s^2, since it's opposite to the initial direction)
s = distance traveled (45.0 m)
v^2 = (1.67 m/s)^2 + 2 * (-2.83 m/s^2) * 45.0 m
v^2 = 2.79 m^2/s^2 + (-254.7 m^2/s^2)
v^2 = -251.9 m^2/s^2
Since the speed cannot be negative, we take the square root of the positive value:
v = √(251.9 m^2/s^2)
v ≈ 15.9 m/s
Therefore, the estimated speed of the rollercoaster when it reaches the bottom of the hill is approximately 15.9 m/s.