Write equation of relation between time period and frequency.

Explanation

From graph, initial velocity `= v _(0)`
and distance travelled in time `t = x _(0).`
For the graph `tan theta = (v_(0))/(x_(0)) = (v _(0) -v)/(x) " "…(1)`
Where, v is velocity and x is displacement at any instant of time t.
From Eq. (i) `v _(0) -v = (v _(0))/(x_(0)) x`
`implies v = (-v _(0))/(x_(0)) x + v _(0) " "...(2)`
We know that

Acceleration `a = (dv)/(dt) = (- v _(0))/( x _(0)) (dx)/(dt) + 0`
`a = (-v_(0))/( x _(0)) (v)`
`= (-v _(0))/( x _(0)) ((-v _(0))/( x _(0)) x + v _(0)) (because `From (2))
`= (v _(0) ^(2))/( x _(0) ^(2)) x - ( v _(0) ^(2))/( x _(0))`
Graph of a versus x is given above.