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:

Answer (Detailed Solution Below)

Option 3 :
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JEE Main 04 April 2024 Shift 1
90 Qs. 300 Marks 180 Mins

Detailed Solution

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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.

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