A motorcar with a mass of 1.3 tons accelerates uniformly from rest up an incline 4% and reaches a speed of 114km/h after 7 minutes. Calculate:
1.The acceleration of a motorcar.
2.the kinetic energy that a motorcar possesses after 7 minutes.
3. The gain in potential energy.
acceleration=change in velocty/time= 114e3/(7*60)(3600) m/s^2
KE=1/2 m v^2=1/2 m (114e3/(7*60))
?? I see that v = 114E3/3600 = 31.667 m/s for (b)
(c) PE = mgh = mg*(1/2 at^2)*sin(arctan(0.04))
To solve these questions, we need to apply relevant formulas from physics. Let's go step by step:
1. The acceleration of a motorcar:
Using the concept of acceleration, we know that acceleration is the change in velocity divided by the time taken. Here, the car starts from rest and achieves a final speed, so the change in velocity would be the final speed.
Given:
Initial Velocity (u) = 0 (car starts from rest)
Final Velocity (v) = 114 km/h
Time taken (t) = 7 minutes = 7 * 60 = 420 seconds
First, we need to convert the final velocity from km/h to m/s:
Speed in m/s = (Speed in km/h) * (1000 m / 3600 s)
v = 114 km/h * (1000 m / 3600 s) = 31.67 m/s
Now, we can use the formula for acceleration:
Acceleration (a) = (Change in Velocity) / (Time taken)
a = (v - u) / t = (31.67 m/s - 0) / 420 s = 0.075 m/s^2
Therefore, the acceleration of the motorcar is 0.075 m/s^2.
2. The kinetic energy that a motorcar possesses after 7 minutes:
The kinetic energy (KE) of an object is given by the formula:
KE = (1/2) * mass * velocity^2
Given:
Mass (m) = 1.3 tons = 1.3 * 1000 kg
Velocity (v) = 31.67 m/s
Substituting the values into the formula, we get:
KE = (1/2) * 1.3 * 1000 kg * (31.67 m/s)^2 = 628,647 J (approximately)
Therefore, the motorcar possesses kinetic energy of approximately 628,647 J after 7 minutes.
3. The gain in potential energy:
The gain in potential energy (PE) is due to the increase in height or altitude. In this case, the car is moving up an incline.
The potential energy gained is given by the formula:
PE = m * g * h
Given:
Mass (m) = 1.3 tons = 1.3 * 1000 kg
Acceleration due to gravity (g) = 9.8 m/s^2 (approximate value)
Incline (h) = 4% => h = (4/100) * length of the incline (h is in meters)
Now we need to calculate the length of the incline. To do that, we need the time taken to reach the final speed.
Using the formula: Distance (d) = (Initial Velocity * Time) + (1/2 * Acceleration * Time^2),
we can calculate the distance (d) traveled by the car in 7 minutes.
Given:
Initial Velocity (u) = 0 (car starts from rest)
Final Velocity (v) = 31.67 m/s
Time taken (t) = 420 seconds
Using the above formula, we can rearrange it to:
Distance (d) = (v * t) + (1/2 * a * t^2)
d = (31.67 m/s * 420 s) + (1/2 * 0.075 m/s^2 * (420 s)^2) = 13,345.25 m (approximately)
Now, we can calculate the height (h):
h = (4/100) * d = (4/100) * 13,345.25 m = 533.81 m (approximately)
Finally, substituting the values into the formula for potential energy, we get:
PE = m * g * h = 1.3 * 1000 kg * 9.8 m/s^2 * 533.81 m = 6,418,199 J (approximately)
Therefore, the gain in potential energy is approximately 6,418,199 J.