A roller coaster reaches the top of the steepest hill with a speed of 3.0 km/h . It then descends the hill, which is at an average angle of 40 degrees and is 50.0 m long. Estimate its speed when it reaches the bottom. Assume kinetic friction (uk) = 0.18.
To estimate the speed of the roller coaster when it reaches the bottom of the hill, we can use the principle of conservation of energy.
First, let's calculate the potential energy of the roller coaster at the top of the hill. The formula for potential energy (PE) is given by:
PE = m * g * h
Where:
m = mass of the roller coaster
g = acceleration due to gravity (approximately 9.8 m/s²)
h = height of the hill
Since we are only given the speed at the top of the hill, we need to calculate the height using the given angle. We can use trigonometry to find the height:
h = L * sin(θ)
Where:
L = length of the hill
θ = angle of the hill (in radians)
Converting the angle from degrees to radians:
θ = 40 degrees * (π/180) ≈ 0.698 radians
Let's substitute the values into the equation:
h = 50.0 m * sin(0.698) ≈ 36.9 m
Now, we can calculate the potential energy at the top of the hill:
PE = m * g * h
Next, as the roller coaster descends the hill, it loses potential energy and gains kinetic energy. At the bottom of the hill, all of the initial potential energy will be converted to kinetic energy, neglecting energy losses due to friction. The formula for kinetic energy (KE) is given by:
KE = 0.5 * m * v²
Where:
v = velocity of the roller coaster at the bottom of the hill
Since energy is conserved, we can equate the potential energy at the top (PE) to the kinetic energy at the bottom (KE):
PE = KE
m * g * h = 0.5 * m * v²
Simplifying and cancelling out the mass:
g * h = 0.5 * v²
Now, let's calculate the speed (v) when the roller coaster reaches the bottom of the hill:
v² = (2 * g * h) / 0.5
v² = (2 * 9.8 m/s² * 36.9 m) / 0.5
v² ≈ 285.24
Taking the square root of both sides:
v ≈ √285.24 ≈ 16.89 m/s
Finally, we can convert the speed from m/s to km/h:
v ≈ 16.89 m/s * (3.6 km/h / 1 m/s) ≈ 60.8 km/h
Therefore, the estimated speed of the roller coaster when it reaches the bottom of the hill is approximately 60.8 km/h.