The coordinate of a particle moving in a plane are given by ` x(t) = a cos (pt) and y(t) = b sin (pt)` where `a,b (lt a)` and `P` are positive constants of appropriate dimensions . Then

Options

A. The path of the particle is an ellipse
B. The velocity and acceleration of the particles are normal to each toerh at t=`pi//2p`.
C. The acceleration of the particle is always directed towards a forucs
D. The distance travelled by the particle in time interval `t=0` to `t=pi//2p` is a.

Answer

a,b

Explanation

`x=alpha cos (pt): y=b sin (pt)`
Equation of path in x-y plane is
`((x)/(a))^(2)+((y)/(b))^(2)=`
i.e., the path of the particle is an elipse.
Position vector of a point P is
`vecr=a cos pt hati+b sin pt hatj`
`rArr vecv=P(-a sin pt hati+b cos pthatj)`
and `veca=-p^(2)(-a cos pt hati+b sin pt hatj)=-p^(2)vecr`
Acceleration of directed towards the centre.
Also, `vecv,veca=0` at `t=pi//2p`
At `t=0,vecr=ahati`
At `t=pi//2p,vecr=bhatj`
So, in time t=0 to `t=pi//2p`, particle goes from A to B travelling a distance more than a.