A classical mechanics problem by Shounak Ghosh
A car starts moving rectilinearly, first with acceleration
w
=5m/s^{2} (initial velocity is equal to zero), then uniformly,then finally,deaccelerating with same rate
w
.The total time of motion
t
=25s.The average velocity during the time equal to
<v>
=72 km/h.How long does car move uniformly?
The answer is 15.
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1 solution
Assume,
Time of motion with acceleration = t = Time of motion with deceleration
Velocity after acceleration, v=u+at=0+at=at
Time of uniform motion = T = 25-2t
Given,
Acceleration, a=5m/s^2
Total time =25s
Average velocity, V=72km/h=20m/s
Now,
Total displacement, s = ( . 5 a t 2 ) + v T + ( . 5 a t 2 ) = a t 2 + a t ( 2 5 − 2 t ) = a t 2 + 2 5 a t − 2 a t 2 = 2 5 a t − a t 2
Then, average velocity V = 2 0 = 2 5 s o r , a t 2 − 2 5 a t + 5 0 0 = 0
Solving this quadratic equation, t=20, 5 . But t=20s is impossible [you know why!] So, t=5 seconds.
Finally, T=25-2t=(25-10)s=15 seconds