A 230-kg roller coaster reaches the top of the steepest hill with a speed of 5.20 km/h. It then descends the hill, which is at an angle of 25° and is 50.0 m long. What will its kinetic energy be when it reaches the bottom? Assume µk = 0.16
I'm not sure how to set up the problem and which equation to use.
To solve this problem, we can use the principles of conservation of mechanical energy and apply the work-energy theorem.
First, we need to find the potential energy of the roller coaster at the top of the hill. The potential energy can be calculated using the formula:
PE = m * g * h
Where:
PE is the potential energy
m is the mass of the roller coaster (230 kg)
g is the acceleration due to gravity (9.8 m/s^2)
h is the height of the hill
Since the roller coaster reaches the top of the hill, its height h is equal to the vertical displacement caused by climbing the hill. We can calculate this using trigonometry:
h = d * sin(θ)
Where:
d is the length of the hill (50.0 m)
θ is the angle of the hill (25°)
Now, we can substitute the values into the formulas:
h = 50.0 m * sin(25°)
h ≈ 21.18 m
PE = 230 kg * 9.8 m/s^2 * 21.18 m
PE ≈ 46855 J
Next, we need to find the kinetic energy of the roller coaster at the bottom of the hill. Since the roller coaster loses some energy due to friction, we need to take into account the coefficient of kinetic friction (µk).
The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy:
Work = ∆KE
The net work done is equal to the sum of the work done by gravity and the work done by friction:
Work = Work_gravity + Work_friction
Work_gravity = PE (potential energy at the top of the hill)
Work_friction = µk * m * g * d (friction force * distance)
Now, let's calculate the work done by friction:
Work_friction = 0.16 * 230 kg * 9.8 m/s^2 * 50.0 m
Work_friction ≈ 17984 J
Substituting the values into the equation:
Work = PE + Work_friction
∆KE = PE + Work_friction
∆KE = 46855 J + 17984 J
∆KE ≈ 64839 J
Therefore, the kinetic energy of the roller coaster when it reaches the bottom of the hill is approximately 64839 J.
To solve this problem, you can use the concepts of potential and kinetic energy, as well as the work-energy theorem. Here's how you can approach it:
Step 1: Convert the speed of the roller coaster from km/h to m/s.
Since the length of the hill is given in meters, it's easier to work with meters per second (m/s) rather than kilometers per hour (km/h). To convert, you can use the conversion factor:
1 km/h = (1,000 m / 1 km) * (1 h / 3600 s) = 0.2778 m/s.
So, the roller coaster's speed at the top of the hill is 5.20 km/h * 0.2778 m/s/km/h = 1.44 m/s.
Step 2: Determine the change in height of the roller coaster.
Since the hill is at an angle, we need to find the vertical displacement or the change in height (Δh) of the roller coaster. To calculate Δh, you can use the given information about the hill's length (50.0 m) and its angle (25°) using trigonometry.
Δh = length of hill * sin(angle)
Δh = 50.0 m * sin(25°)
Step 3: Calculate the change in potential energy of the roller coaster.
The change in potential energy (ΔPE) is equal to the product of the mass (m) and the change in height (Δh) multiplied by the acceleration due to gravity (g = 9.8 m/s^2).
ΔPE = m * Δh * g
Note: In this case, the initial height is considered the top of the hill, and the final height is considered the bottom of the hill.
Step 4: Calculate the work done against friction.
The work (W) done against friction can be calculated using the equation:
W = force of friction * distance
To find the force of friction, you can use the coefficient of kinetic friction (µk) and the normal force (Fn). The normal force is equal to the weight (m * g) of the roller coaster, where g is the acceleration due to gravity.
force of friction = µk * Fn
force of friction = µk * (m * g)
Step 5: Apply the work-energy theorem.
The work-energy theorem states that the net work done on an object is equal to its change in kinetic energy (ΔKE). In this case, the roller coaster starts with zero kinetic energy (KE) at the top of the hill and ends up with some kinetic energy at the bottom of the hill.
ΔKE = W + ΔPE
Step 6: Substitute the values into the equation and solve for KE.
The kinetic energy (KE) at the bottom of the hill can be found by rearranging the equation:
KE = ΔKE - ΔPE
By following these steps and plugging in the appropriate values, you can find the roller coaster's kinetic energy when it reaches the bottom of the hill.