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The coordinate of a particle moving in ...

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

A

The path of the particle is an ellipse

B

The velocity and acceleration of the particle are normal to each other at `t=pi//2p`

C

The acceleration of the particle is always directed towards a focus.

D

The distance travelled by the particle in time interval `t=0` to `t=pi/(2p)` is a

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
`x=a cospt, y=b sin pt, vec(r)=a cos (pt) hat(i)+b sin (pt) hat(j)`
`:' sin^(2)pt+cos^(2)pt=1`
`:. x^(2)/a^(2)+y^(2)/b^(2)=1` (ellipse)
`vec(v)=-ap sin (pt)hat(i)+bp cos (pt)hat(j), v_(t)=pi/(2p)=-aphat(i)`
`vec(a)=-ap^(2)(pt)hat(i)=bp^(2) sin (pt)hat(j), a_(t)=pi/(2p)=-bp^(2)hat(j)`
`vec(a).vec(v)=0`
`vec(a)=-p^(2) [a cos pthat(i)+b sin pthat(j)]=-p^(2)vec(r)`
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Knowledge Check

  • The coordinates of a particle moving in a plane are given by x(t)=acos(pt)andy(t)=bsin(pt)" where "a,b(lta)andp are positive constants of appropriate dimensions. Then

    A
    the path of the particle is an ellipse.
    B
    the velocity and acceleration of the particle are normal to each other at `t=pi//(2p)`.
    C
    the acceleration of the particle is always directed toward a focus.
    D
    the distance traveled by the particle in time interval t = 0 to `t=pi//(2p)` is a.

  • The coordinates of a moving particle at any time t are given by x = ct and y = bt. The speed of the particle at time t is given by

    A
    2t(c-b)
    B
    `sqrt((c^(2)+b^(2)))`
    C
    `2tsqrt((c^(2)+b^(2)))`
    D
    `2tsqrt((c^(2)+b^(2)))`

  • The coordinates of a moving particle at any time t are given by x = ct and y = bt^(2) . The speed of the particle is given by

    A
    `2t sqrt(b^(2)-c^(2))`
    B
    `sqrt(4b^(2)t^(2)+c^(2))`
    C
    `2t(b+c)`
    D
    `2t(b-c)`

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