A roller coaster reaches the top of the steepest hill with a speed of 7.4 km/h. It then descends the hill, which is at an average angle of 43° and is 45.0 m long. What will its speed be when it reaches the bottom? Assume µk = 0.12.
____m/s
Any help appreciated :)
To find the speed of the roller coaster when it reaches the bottom of the hill, we need to consider the conservation of mechanical energy. At the top of the hill, the roller coaster has potential energy (due to its height) and kinetic energy (due to its velocity).
First, let's convert the initial speed from kilometers per hour (km/h) to meters per second (m/s) since the rest of the problem uses SI units.
Given:
Initial speed at the top of the hill, v_0 = 7.4 km/h
To convert km/h to m/s, we can use the following conversion factor: 1 km/h = 0.2778 m/s
v_0 = 7.4 km/h * 0.2778 m/s per km/h
v_0 = 7.4 m/s
Now, let's find the potential energy at the top of the hill. The formula for potential energy is given by:
Potential energy (PE) = mass (m) * acceleration due to gravity (g) * height (h)
Since the problem doesn't specify the mass of the roller coaster, we can cancel it out by dividing both sides of the equation by m:
PE/m = g * h
The mass cancels out, leaving us with:
PE = mgh
Where:
PE = potential energy
m = mass
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height
Next, let's find the height of the hill using the given information. Since the hill is at an average angle of 43 degrees and is 45.0 meters long, we can use trigonometry.
The height (h) of the hill is given by:
h = length of the hill * sin(angle)
h = 45.0 m * sin(43°)
Use a calculator to find the value of sin(43°) and multiply it by 45.0 m to get the height.
h ≈ 45.0 m * 0.682 ≈ 30.69 m
Now that we know the height of the hill, we can find the potential energy at the top using the formula:
PE = mgh
PE = (mass of the roller coaster) * 9.8 m/s^2 * 30.69 m
Next, since energy is conserved, the initial potential energy will be converted into kinetic energy at the bottom of the hill. Therefore, the kinetic energy at the bottom is equal to the potential energy at the top.
Kinetic energy (KE) = (mass of the roller coaster) * (velocity at the bottom)^2 / 2
Setting the potential energy equal to the kinetic energy:
PE = KE
(mass of the roller coaster) * 9.8 m/s^2 * 30.69 m = (mass of the roller coaster) * (velocity at the bottom)^2 / 2
Simplify the equation by canceling the mass of the roller coaster:
9.8 m/s^2 * 30.69 m = (velocity at the bottom)^2 / 2
Solve for (velocity at the bottom)^2:
(velocity at the bottom)^2 = 2 * 9.8 m/s^2 * 30.69 m
Finally, take the square root of both sides to find the velocity at the bottom:
velocity at the bottom = √(2 * 9.8 m/s^2 * 30.69 m)
Using a calculator, evaluate the expression to find the final velocity in m/s.
So, the final velocity when the roller coaster reaches the bottom of the hill will be ______ m/s.