A partile starts from origin at t = 0 with a velocity `5.0 hat(i)` m/s and moves in x-y plane under action of a force which produces a constant acceleration of `(3.0 hat(i) + 2.0 hat(j)) m//s^(1)`. (a) What is the y-coordinate of the particle at the instant its x-coordinate i 84 m? (b) What is the speed of the particle at this time ?

Explanation

The position of the particle is given by
`r(t) = v_(0)t+(1)/(2) a t^(2)`
`= 5.0 hat(i) t + (1//2)(3.0 hat(i) + 2.0 hat(j))t^(2)`
`= (5.0 t + 1.5 t^(2)) hat(i) + 1.0 t^(2)hat(j)`
Therefore x(t) `= 5.0 t + 1.5 t^(2)`
`y(t) = + 1.0 t^(2)`
Given `x(t) = 84m , t = ?`
`5.0 t + 1.5 t^(2) = 84 rArr t = 6s`
At t = 6 s, `y = 1.0 (6)^(2) = 36.0 m`
Now, the velocity `v = (d r)/(d t)`
`= (5.0 + 3.0t) hat(i) + 2.0 t hat(j)`
At t = 6s, v `= 23.0 hat(i) + 12.0 hat(j)`
speed `= |v| = sqrt(23^(2) + 12^(2)) ~= 26 ms^(-1)`