How much work must be done to accelerate an 8x105 kg train: a) from 10 m/s to 15 m/s; b) from 15 m/s to 20 m/s; c) to a stop an initial speed of 20 m/s?
To calculate the work done to accelerate an object, we can use the equation:
Work (W) = Force (F) * Distance (d)
But first, we need to calculate the force required to accelerate the train:
Force (F) = mass (m) * acceleration (a)
Given:
mass (m) = 8x10^5 kg
(a) From 10 m/s to 15 m/s:
Initial velocity (u) = 10 m/s
Final velocity (v) = 15 m/s
Step 1: Calculate the acceleration (a):
a = (v - u) / t
If the time (t) is not given, we can assume it to be 1 second.
a = (15 - 10) / 1 = 5 m/s^2
Step 2: Calculate the force (F):
F = m * a
F = 8x10^5 kg * 5 m/s^2 = 4x10^6 N
Step 3: Calculate the work (W):
W = F * d
Since the distance (d) is not given, we cannot calculate the exact work done. However, we can say that work is equal to the force multiplied by the distance over which it acts. Hence, the value of work will depend on the distance traveled by the train during acceleration.
(b) From 15 m/s to 20 m/s:
Initial velocity (u) = 15 m/s
Final velocity (v) = 20 m/s
Step 1: Calculate the acceleration (a):
a = (v - u) / t
Assuming the time (t) to be 1 second again:
a = (20 - 15) / 1 = 5 m/s^2
Step 2: Calculate the force (F):
F = m * a
F = 8x10^5 kg * 5 m/s^2 = 4x10^6 N
Step 3: Calculate the work (W):
W = F * d
Similar to the previous case, we need the distance (d) to calculate the exact work done.
(c) To stop from an initial speed of 20 m/s:
Initial velocity (u) = 20 m/s
Final velocity (v) = 0 m/s
Step 1: Calculate the deceleration (a):
a = (v - u) / t
Again, assuming the time (t) to be 1 second:
a = (0 - 20) / 1 = -20 m/s^2
Here, the deceleration is negative since it is opposing the motion.
Step 2: Calculate the force (F):
F = m * a
F = 8x10^5 kg * (-20 m/s^2) = -1.6x10^7 N
Step 3: Calculate the work (W):
W = F * d
As mentioned earlier, we need the distance (d) to calculate the exact work done.
In conclusion, we can determine the force required for acceleration and deceleration, but the exact work done depends on the distance traveled by the train in each case.
To calculate the work done to accelerate an object, we can use the formula:
Work = Force * Distance,
where Force is the net force acting on the object and Distance is the distance over which the force is applied.
First, let's determine the force required to accelerate the train. The net force can be determined using Newton's second law:
Force = Mass * Acceleration,
where Mass is the mass of the train and Acceleration is the change in velocity divided by the time taken.
a) From 10 m/s to 15 m/s:
Since the train is accelerating, we need to find the average acceleration first. Given the final velocity (15 m/s) and the initial velocity (10 m/s), the change in velocity is 15 m/s - 10 m/s = 5 m/s.
If the time taken to achieve this acceleration is not mentioned, we can assume it to be one second for simplicity.
Using the formula, Acceleration = Change in velocity / Time taken, we get:
Acceleration = 5 m/s / 1 s = 5 m/s^2.
Now, we can calculate the force required using Newton's second law:
Force = Mass * Acceleration = (8x10^5 kg) * (5 m/s^2) = 4x10^6 N.
To calculate the work done, we need to know the distance over which this force is applied. Unfortunately, this information is missing in the question, so we cannot determine the exact work done.
b) From 15 m/s to 20 m/s:
Using the same approach, let's calculate the force required.
Change in velocity = 20 m/s - 15 m/s = 5 m/s.
Acceleration = Change in velocity / Time taken = 5 m/s / 1 s = 5 m/s^2.
Force = Mass * Acceleration = (8x10^5 kg) * (5 m/s^2) = 4x10^6 N.
Again, without the distance over which the force is applied, we cannot determine the exact work done.
c) To a stop from an initial speed of 20 m/s:
The train is coming to a stop, so the final velocity is zero.
Change in velocity = 20 m/s - 0 m/s = 20 m/s.
Acceleration = Change in velocity / Time taken = 20 m/s / 1 s = 20 m/s^2.
Force = Mass * Acceleration = (8x10^5 kg) * (20 m/s^2) = 1.6x10^7 N.
Without the distance, we cannot calculate the work done.
In summary, to calculate the work done, we need to know the distance over which the force is applied.