If a train starts from rest and accelerate at the rate alpha and attains velocity v ,then moves at a constant velocity,then deaccelerates at the rate beta and stops ,if the total distance is found to be l then what is the total time??
To find the total time taken by the train to cover the total distance l, we need to break down the problem into three parts: the initial acceleration, the constant velocity, and the deceleration.
Let's consider each part separately:
1. Initial Acceleration:
When the train starts from rest and accelerates at the rate alpha, we can use the formula of motion:
v = u + at,
where:
v = final velocity (given as v),
u = initial velocity (which is 0 since the train starts from rest),
a = acceleration (which is given as alpha), and
t = time taken for initial acceleration.
Simplifying the equation, we get:
v = 0 + alpha * t,
t = v / alpha.
2. Constant Velocity:
Once the train reaches the velocity v, it moves at a constant velocity. Let's denote the time taken during this phase as t_const.
Since the velocity is constant, the distance covered during this phase can be calculated using the formula:
Distance = Velocity * Time,
l_const = v * t_const.
3. Deceleration:
After the constant velocity phase, the train decelerates at a rate of beta until it comes to a stop. The deceleration is negative, so we use the same formula as in the initial acceleration:
v = u + at,
where:
v = final velocity (which is 0 since the train stops),
u = initial velocity (which is v, the constant velocity),
a = deceleration (which is negative beta), and
t = time taken to decelerate.
Simplifying the equation, we get:
0 = v - beta * t,
t = v / beta.
Now, we can calculate the total time taken by adding up the time taken for each phase:
Total Time = t + t_const + t_deceleration,
Total Time = v / alpha + t_const + v / beta.
Therefore, the total time taken for the train to cover the total distance l can be given by the equation:
Total Time = v / alpha + l_const / v + v / beta.