Question

A particle moving in the $xy$ plane experiences a velocity-dependent force, 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?

• A. The kinetic energy of the particle is constant in time.
• B. quantity $v\times a$ is constant in time
• C. quantity $v.a$ is constant in time
• D. $F$ arises due to a magnetic field

Answer 1

The correct option is B

quantity $v\times a$ is constant in time

The explanation for the correct option(s):

Step 1. Check Option (B) quantity is constant in time.

We have, Velocity-dependent-force along $XY$ plane.

$F=k\left(v_x\hat{i}+v_y\hat{j}\right)$ ……$\left(i\right)$

Where $k$ is proportionally constant, $F$ is the force, $v_x$ is the motion along $X$ component, and $v_y$ is the motion along $Y$ component .

Also,

$F=ma$ …..$\left(ii\right)$

Where $m$ is the mass and $a$ is the acceleration,

From equations $\left(i\right)$ and $\left(ii\right)$ we get.

$ma=k\left(v_x\hat{i}+v_y\hat{j}\right)$

Now,

$a_x=\frac{k{v}_y}{m}$ and

$a_y=\frac{k{v}_x}{m}$

$a=\frac{kv}{m}$

Therefore from the above relation, it is clear that $v$ and $a$ is in the same direction.

Hence, $v\times a=0$ and constant in time.

The explanation for the incorrect option(s):

Step 2. Check Option (A) The kinetic energy of the particle is constant in time.

1. The given force is not a central force. So work has to be done by this force and it will bring a change in kinetic energy.
2. Hence, it is incorrect.

Step 3. Check Option (C) quantity v . a is constant in time.

1. Since, $v.a=\left(v_x\hat{i}+v_y\hat{j}\right)\times \frac{k}{m}\left(v_y\hat{i}+v_x\hat{j}\right)=\frac{2k}{m}{v}_x{v}_y$
2. Which is not constant as force and is not zero.
3. Hence, it is incorrect.

Step 4. Check Option (D) F arises due to a magnetic field.

1. Magnetic force can never change the speed of a charged particle. So force (F) does not arise due to the magnetic field.
2. Hence, it is incorrect.

Hence, option $\left(B\right)$ is correct.