Question

Problem 5: The greatest possible acceleration or deceleration that a train may have is a and its maximum speed is v. Find the minimum time in which the train can get from one station to the next if the total distance between the stations is s. Assume the train starts from the first station at rest and stops when it reaches the next station.

Answer 1

The acceleration of the train is not uniform, therefore, for t₁ time train moved with 'a' acceleration and t ₂ time it moved with uniform acceleration, here it is zero and t₃ time it moved with deceleration 'a ' that is total time is given as

t_min= t₁ + t₂ + t₃

Then, with 'v' as maximum speed

$t_{min}= \frac{v}{a} +0+\frac{v}{a} ..............(1)$

$t_{min}= 2\frac{v}{a} .............(2)$

We know,

v= at

Therefore,

$S= \frac{1}{2} a t_{min}^{2} = \frac{1}{2} v t_{min} ..............(3)$

where S is the distance between the two stations   

Eliminating v from eqn.(3) using eqn.(2), we get

$S= \frac{a t_{min}^{2}}{4}$

Therefore ,

$t_{min}=\sqrt{\frac{4S}{a}};$

$\mathbf{t_{min}=2\sqrt{\frac{S}{a}} }$