Question

A particle projected from origin moves in x-y plane with a velocity v=3i+6xj where I and j are unit vectors along x andvy axis.What is the equation of path followed by particle?

Answer 1

`x = 3t+x_0, y = 9t^2 +6x_0 t+y_0`

Explanation:

Since `vec{v} = {dx}/{dt } i +{dy}/{dt}j` we get

`dx/dt = 3` and `dy/dt = 6x`

The first of these is easily solved to give

`x = 3t+x_0`

Substituting this in the second equation leads to

`dy/dt = 18t+6x_0`

which can be easily integrated to give

`y = 9t^2 +6x_0 t+y_0`

(here we have assumed that the initial position is `(x_0 ,y_0)`

Thus, the equation of the path is given by the pair of equations `x = 3t+x_0, y = 9t^2 +6x_0 t+y_0`. While this is a perfectly good parametric representation of the path, we may easily eliminate the parameter `t` to write the quation in a more familiar form.