A particle moves in XY-plane with x and y varying with time t as x(t) = 5t, y(t) = `5t (27 -t^(2))`. At what time in seconds, the direction of velocity and acceleration will be perpendicular to each other ?

Options

A. `5 sqrt((27)/(2))`
B. 5
C. `5 sqrt12`
D. 3

Answer

D

Explanation

Velocity in x-direction is
`v_(x) = (d)/(dt) (x) = (d)/(dt) (5t) = 5 ms^(-1)`
velocity in y-direction is
`v_(y) = (d)/(dt) (y) = (d)/(dt) (5 xx 27t - 5t^(3)) = 5 xx 27 - 5 xx 3t^(2)`
As, `v_(x) =` constant and `v_(y)` = time dependent.
`:.` This is a case similar to projectile motion and velocity and acceleration are perpendicular, when `v_(y) = 0`.
`rArr 5 xx 27 - 5 xx 3t^(2) = 0 rArr t^(2) = 9 rArr t = 3s`