The acceleration (a)-time(t) graph for a particle moving along a straight from rest is shown in figur. Which of the following graph is the best representation of its velocity (v) with time (t) ?

Options

A.
B.
C.
D.

Answer

A

Explanation

Given acceleration-time graph is shown in figure.

Let `a(t=0)=a_(0)=` constant
From the graph, we have
`(t)/(T)+(a)/(a_(0))=1`
`rArr a_(0)t+aT=a_(0)T` ….(i)
Also, we know that
`a=(d upsilon)/(dt)` ....(ii)
From Eqs. (i) and (ii), we have
`(a_(0)T-a_(0)t)/(T)=(d upsilon)/(dt)`
Integrating both sides, we get
`int_(0)^(t)(a_(0)T-a_(0)t)dt=T int_(0)^(upsilon)d upsilon`
`rArr a_(0)Tt-(a_(0)t^(2))/(2)=T upsilon rArr = a_(0)t-(a_(0))/(2T)t^(2)`
Therefore, `upsilon-t` graph must be parabolic.
Now, at `t = 0, upsilon = 0`
`t=T, upsilon=(a_(0)T)/(2)` = constant
These conditions are asatisfied by graph (a).