The tracks of Angel's flight extend 96.0 m along the side of a hill at an angle of 18.4 degrees with respect to the horizontal. If each car has a passenger with a mass of 70.0 kg, what is the total mechanical energy associated with the two passengers when the cars are about to leave the boarding platform?
I don't understand the problem.
To find the total mechanical energy associated with the two passengers on the Angel's Flight, we need to consider the potential energy and kinetic energy.
1. Potential Energy:
The potential energy is given by the formula: PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the vertical height.
Since the track is along the side of the hill, we need to find the vertical height component:
h = d * sin(theta)
where d is the length of the tracks and theta is the angle with respect to the horizontal.
Let's calculate h:
d = 96.0 m
theta = 18.4 degrees
Using the sine function:
h = 96.0 * sin(18.4)
2. Kinetic Energy:
The kinetic energy is given by the formula: KE = (1/2)mv^2, where m is the mass and v is the velocity.
Since the cars are about to leave the boarding platform, they are not moving. Therefore, their kinetic energy is zero.
Now, let's calculate the total mechanical energy:
Total Mechanical Energy = Potential Energy + Kinetic Energy
Total Mechanical Energy = PE + KE
Total Mechanical Energy = mgh + 0
Substituting the values:
Total Mechanical Energy = (70.0 kg) * (9.8 m/s^2) * (96.0 m * sin(18.4)) + 0
Calculate the value to get the total mechanical energy.
To find the total mechanical energy associated with the two passengers, we first need to calculate the gravitational potential energy and the kinetic energy for each passenger. Then, we can add up these energies to get the total mechanical energy.
Let's start by calculating the gravitational potential energy for one passenger. The formula for gravitational potential energy is:
PE = m * g * h
Where:
- PE is the gravitational potential energy
- m is the mass of the passenger
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- h is the height above a reference point
In this case, the height h is the vertical displacement of the passenger as they travel along the hill. To determine this height, we can use the given angle of 18.4 degrees and the length of the track along the hill.
The vertical displacement can be found by calculating:
h = track length * sin(angle)
Substituting the values:
h = 96.0 m * sin(18.4 degrees)
Now we can calculate the gravitational potential energy for one passenger:
PE = 70.0 kg * 9.8 m/s^2 * h
Next, let's calculate the kinetic energy for one passenger. The formula for kinetic energy is:
KE = 1/2 * m * v^2
Where:
- KE is the kinetic energy
- m is the mass of the passenger
- v is the velocity of the passenger
The velocity of the passenger can be found using the horizontal length of the track and the time it takes for the passenger to travel that distance.
To find the time, we assume that the ride is frictionless and neglect any other forces. Therefore, only gravity will be providing the acceleration. The equation for the horizontal displacement can be determined using the formula:
d = v * t
Solving for t:
t = d / v
In this case, the horizontal length of the track is same as the length along the hill, which is 96.0 m. So:
t = 96.0 m / v
Now, to find the velocity v, we need to use the horizontal component of the passenger's acceleration, which is equal to the acceleration due to gravity multiplied by the cosine of the angle.
So, the velocity can be calculated as:
v = a * t
v = g * cos(angle) * t
Substituting the known values:
v = 9.8 m/s^2 * cos(18.4 degrees) * (96.0 m / v)
Now we can solve this equation to find v.
After finding the velocity, we can calculate the kinetic energy:
KE = 1/2 * 70.0 kg * v^2
Finally, to find the total mechanical energy, we add up the gravitational potential energy and the kinetic energy for one passenger:
Total mechanical energy = PE + KE + PE + KE
Remember to substitute the calculated values for h and v into the equations before adding them up.