Question

“It is impossible for a particle to move in a straight line so that its velocity varies at the distance from the commencement of motion”. Which one is correct for the given statement?

(a) The above statement is valid

(b) The above statement is not valid

(c) Data inadequate

(d) Answer does not exist

This question was addressed to me in an online interview.

My query is from Calculus Application in division Application of Calculus of Mathematics – Class 12

Answer 1

The correct option is (a) The above statement is valid

For explanation: Let a particle be moving along the straight-line OX and P be its position at time t, when OP = x. Clearly, the velocity and acceleration of the particle at P are dx/dt and d^2x/dt^2 respectively.

If possible, let us assume,

dx/dt α x

dx/dt = kx, where k is a constant variation.

Now, d^2x/dt^2 = d(kx)/dt = k(dx/dt)

= k^2x

At the starting point O, we have x = 0 and we see that dx/dt = 0 and d^2x/dt^2 = 0, when x = 0 that is, both velocity and acceleration of the particle are zero at x = 0 and the particle remains at rest at O.

Therefore, it is impossible for a particle to have any motion under the given condition.