JEE Main · 2020 · 06 Sep 2020 · Shift 2 · Work, Power & Energy

A particle moving in the

x y

plane experiences a velocity dependent force

F = k (v_{y} \hat{i} + v_{x} \hat{j})

, where

v_{x}

and

v_{y}

are the

x

and

y

components of its velocity

v

. If

a

is the acceleration of the particle, then which of the following statements is true for the particle?

Answer: (A)

Solution

The force splits into components

m a_{x} = k v_{y}

and

m a_{y} = k v_{x}

, so the acceleration is

a = \frac{k}{m} (v_{y} \hat{i} + v_{x} \hat{j})

. Dividing the two component equations gives

\frac{d v _{y}}{d v _{x}} = \frac{v _{x}}{v _{y}}

, which on integration yields

v_{y}^{2} - v_{x}^{2} =

constant. Now look at

v \times a = (v_{x} \hat{i} + v_{y} \hat{j}) \times \frac{k}{m} (v_{y} \hat{i} + v_{x} \hat{j}) = \frac{k}{m} (v_{x}^{2} - v_{y}^{2}) \hat{k}

. Since

v_{x}^{2} - v_{y}^{2}

stays fixed in time, the vector

v \times a

keeps a constant value throughout the motion.

JEE Main · 2020 · 06 Sep 2020 · Shift 2 · Single correct option