A particle is projected with velocity v₀ along x-axis. A damping force is acting on the particle which is proportional to the square of the distance from the origin i.e. ma = -ax². The distance at which the particle stops:

CONCEPT:

• The damping force is defined as the force which is proportional to the velocity of the mass but opposite to the motion of the mass.
• Newton's second law of motion, states that the force which is equal to the rate of change of momentum. For a constant mass, force is equal to mass times acceleration.

         F = ma

Here, F is the force, m is the mass and a is the acceleration of the given body.

CALCULATION:

It is given that the damping force of the particle is proportional to the square of the distance and it is written as,

F =- x²      ----(1)

According to Newton's law, we have;

F = ma        ----(2)
Now, from equation (1) and equation (2) we have;

ma = - x²

⇒  

⇒ 

⇒      ----(3)

Now, Integrating equation (3) we have;

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

Hence option 1) is the correct answer.