Two cars P and Q start from a point at the same time in a straight line and their positions are represented by `x_(p) (t) = at + bt ^(2) and x _(Q) (t) = f t - t ^(2).` At what time do the cars have the same velocity?

Options

A. `(a+f)/(2 (a+b))`
B. `(f-a)/(2(1 +b))`
C. `(a-f)/(a +b)`
D. `(a+f)/(2 (b-1))`

Answer

B

Explanation

`x_(p) (t) = at + bt ^(2) x _(Q) (t) ft - t ^(2)`
`therefore u _(P) = (d)/(dt) (at+ bt ^(2)) therefore v _(A) = (d)/(dt) (ft -t ^(2))`
`therefore u _(P) =a + 2bt therefore v _(Q) = f - 2t`
But ` v _(P) =v_(Q)`
`therefore a + 2 bt = f - 2t`
`therefore 2bt + 2t =f -a`
`therefore 2t) b +1) =f -a`
`therefore t = (f-a)/(2 (1 + b))`