Question

The velocity of a train increases at a constant rate alpha from 0m/sec to vm/sec and then remains constant for some time interval and then finally decreases to 0 at a constant rate beta. if the total distance covered by the particle is x then,prove that the total time taken will be t=x/v+v/2[1/alpha+1/beta]​

Answer 1

Answer:

v=αt

1

⇒t

1

=

α

v

v=βt

1

⇒t

2

=

β

v

∴t

0

=t−t

1

−t

2

=(t−

α

v

−

β

v

)

Now,l=

2

1

αt

1

2

+vt

0

+

2

1

βt

2

1

⇒

2

1

α(

α

v

)

2

+v(t−

α

v

−

β

v

)+

2

1

β(

β

v

)

2

⇒vt−

2α

v

2

−

2β

v

2

⇒t=

v

l

+

2

v

(

α

1

+

β

1

)

For t to be minimum its first derivative with respect to velocity should be zero or,

⇒0=−

v

2

l

+

2αβ

α+β

⇒v=

α+β

2lαβ