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Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A): The motion of a particle described by the position vector r = A ( t) i + B ( t) j (where A B ) is an elliptical trajectory, and the net force acting on the particle is always directed towards one of the foci of the ellipse. Reason (R): The acceleration of the particle satisfies the relation a = - ^

Options

  1. ABoth (A) and (R) are true and (R) is the correct explanation of (A).
  2. BBoth (A) and (R) are true but (R) is not the correct explanation of (A).
  3. C(A) is true but (R) is false.
  4. D(A) is false but (R) is true.

Correct answer

D. (A) is false but (R) is true.

Step-by-step solution

For the given position vector r = A ( t) i + B ( t) j , the coordinates are x = A ( t) and y = B ( t) . Eliminating t , we get x^2 A^2 + y^2 B^2 = 1 , which is the equation of an ellipse. Thus, the trajectory is elliptical. The acceleration is given by a = d^2 r dt^2 = - ^2 [A ( t) i + B ( t) j ] = - ^2 r . According to Newton's second law, the net force is F = m a = -m ^2 r . The negative sign indicates that the force is always directed opposite to the position vector, i.e., towards the origin (the center of the e

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