A vehicle with mass of 1200kg accelerates from rest upward with an incline of 1:25 and reaches a speed of 72km/h after 2 minutes, calculate the potential energy
how high
average speed = 72/2*10^3 = 36 *1000 meters / (3600 seconds) = 10 m/s
distance up = h = 10 m/s * 120 s * (1/25) = 48 meters
PE = U = m g h = 1200 * 9.81 on earth * 48 Joules
To calculate the potential energy, we need to find the height the vehicle has gained.
First, let's convert the speed from km/h to m/s:
72 km/h = (72 * 1000) m/ (60 * 60) s = 20 m/s
Now, we can use the formula for potential energy:
Potential energy (PE) = mass (m) * gravity (g) * height (h)
Given:
mass (m) = 1200 kg
gravity (g) = 9.8 m/s^2
We need to find height (h).
The incline is given as 1:25. This means that for every 1 meter of vertical height, there is a 25-meter increase in horizontal distance.
Let's assume the vertical height gained is x meters.
Then, the horizontal distance gained will be 25x meters.
We can use the formula for acceleration on an inclined plane:
acceleration (a) = g * sin(theta)
where theta is the angle of the incline.
Given:
theta = arctan (1/25)
Now, let's find the acceleration:
a = 9.8 * sin(theta)
Finally, using the kinematic equation to find height h, with an initial velocity of 0 m/s and acceleration a over a time of 2 minutes (converted to seconds):
h = 0.5 * a * t^2
where t = 2 * 60 = 120 seconds
Now, let's calculate the potential energy using the formula:
PE = m * g * h
Substituting the values:
PE = 1200 * 9.8 * h
Now, let's calculate each step:
Step 1: Convert speed from km/h to m/s
72 km/h = 20 m/s
Step 2: Calculate the inclination angle
theta = arctan (1/25)
Step 3: Calculate the acceleration
a = 9.8 * sin(theta)
Step 4: Calculate the height gained
h = 0.5 * a * t^2 = 0.5 * a * (120^2)
Step 5: Calculate the potential energy
PE = 1200 * 9.8 * h
Now, let's compute these steps:
Step 1: 72 km/h = 20 m/s
Step 2: theta = arctan (1/25) ≈ 2.29 degrees
Step 3: a = 9.8 * sin(theta) ≈ 0.378 m/s^2
Step 4: h = 0.5 * a * (120^2) = 0.5 * 0.378 * 14400 ≈ 2721.6 m
Step 5: PE = 1200 * 9.8 * 2721.6 ≈ 31,905,984 J
Therefore, the potential energy of the vehicle is approximately 31,905,984 Joules.