When a particle is moving in a plane it can be convenient to consider separately its motion in two perpendicular directions. If displacement, velocity and acceleration are functions of time, then the calculus methods used so far can be applied to the components in each direction.

Consider, for example, the motion of the particle shown in the diagram.

 

 

 

 

 

 

 

 

In the direction Ox, at time t, x =

i.e. the displacement from O is

therefore, the velocity is 2t

and the acceleration is 2

In the direction Oy, y =

i.e. the displacement from O is

therefore, the velocity is

and the acceleration is 6t

Now we can express these components in terms of unit vectors i and j in the chosen direction and, by adding them, form a resultant vector.

For example, the position vector of P is denoted by r, and is given by

r = i + ( + t)j

Similarly the velocity vector, v is 2ti + ( )j and the acceleration vector, a, is 2i + 6tj

As each component of v is obtained by differentiating the corresponding component of s with respect to t, we can say

and

Similarly,

and