Question

Derive equations of motion for a particle
moving in a plane and show that the
motion can be resolved in two independent
motions in mutually perpendicular
directions. ​

Answer 1

Answer:

Displacement in 2-D plane begin mathsize 14px style S with rightwards arrow on top space equals space S subscript x space i with hat on top space plus space S subscript y space j with hat on top space end style

Velocity in 2D-plane begin mathsize 14px style V with rightwards arrow on top space equals space V subscript x space i with hat on top space plus space V subscript y space j with hat on top space end style

Acceleration in 2D-plane begin mathsize 14px style a with rightwards arrow on top space equals space a subscript x space i with hat on top space plus space a subscript y space j with hat on top space end style

where i and j are unit vectors along x-axis and y-axis of cartesian coordinate system.

For uniform accelerated motion we have 3 equations of motion in vector form

begin mathsize 14px style v with rightwards arrow on top space equals space u with rightwards arrow on top space plus space a with rightwards arrow on top space t space space................ left parenthesis 1 right parenthesis

S with rightwards arrow on top space equals space u with rightwards arrow on top t space plus space 1 half space a with rightwards arrow on top space t squared space space.............. left parenthesis 2 right parenthesis

v with rightwards arrow on top times v with rightwards arrow on top space equals space u with rightwards arrow on top times u with rightwards arrow on top space plus space 2 space a with rightwards arrow on top times S with rightwards arrow on top space space.......... left parenthesis 3 right parenthesis end style

If we choose a cartesian coordinate system, motion in any direction can be resolved in x-direction and y-direction.

Resolved form of equations (1), (2) and (3) are used to describe the motion individually in x and y directions.

Then resultant of resolved component will be considered to get net effect.

For example, in x-direction, resolved form of eqn.(1) along x-direction, i.e., " vx = ux + ax t "

can be used to find the relation among initial velocity ux along x-direction, final velocity vx along y-direction,

acceleration ax along x-direction and time duration t