A particle of mass m is initially at rest at the origin. It is subjected to a force and starts moving along the x-axis. Its kinetic energy K changes with time as dK/dt = γt, where γ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?a)The force applied on the particle is constantb)The speed of the particle is proportional to timec)The distance of the particle from the origin increases linearly with timed)The force is conservativeCorrect answer is option 'A,B,D'. Can you explain this answer?

Question

A particle of mass m is initially at rest at the origin. It is subjected to a force and starts moving along the x-axis. Its kinetic energy K changes with time as dK/dt = &gamma;t, where &gamma; is a positive constant of appropriate dimensions. Which of the following statements is (are) true?a)The force applied on the particle is constantb)The speed of the particle is proportional to timec)The distance of the particle from the origin increases linearly with timed)The force is conservativeCorrect answer is option 'A,B,D'. Can you explain this answer?

Answer

The kinetic energy of the particle is given by,

Integrate both sides of the above equation.

As γ and m are constant; therefore, v ∝t.
The acceleration of the particle is calculated as,

The force on the particle is given by,
F = ma

F = constant
Find the distance of the particle from the origin.

Thus, distance does not increase linearly with time.

Answer

Answer

Answer

The equation dK/dt represents the rate of change of kinetic energy with respect to time. In this case, we are given that the particle is subjected to a force, which means that there is an external force acting on the particle.

According to Newton's second law of motion, the force acting on an object is equal to the rate of change of its momentum. Since momentum is defined as the product of mass and velocity (p = mv), we can write the force as F = dp/dt = d(mv)/dt.

Using the chain rule of differentiation, we can expand the expression for force as:

F = d(mv)/dt = m(dv/dt) + v(dm/dt)

Since the particle is initially at rest, its velocity is zero (v = 0), so the second term in the equation becomes zero.

Therefore, the force acting on the particle is simply F = m(dv/dt).

Now, we know from the work-energy theorem that the work done on an object is equal to the change in its kinetic energy. Mathematically, this can be expressed as:

W = ∫F dx = ∫m(dv/dt) dx

Since the particle is moving along the x-axis, we can rewrite dx as v dt:

W = ∫m(dv/dt) v dt

Integrating both sides of the equation, we get:

∫W dt = ∫m v dv

The left-hand side of the equation represents the change in kinetic energy, which we can represent as ∆K. Therefore, the equation becomes:

∆K = ∫m v dv

Integrating the right-hand side of the equation, we get:

∆K = (1/2) m v^2

Therefore, the rate of change of kinetic energy with respect to time is given by:

dK/dt = d((1/2) m v^2)/dt = m v (dv/dt)

In summary, the rate of change of kinetic energy with respect to time is given by dK/dt = m v (dv/dt), where v is the velocity of the particle and dv/dt is the acceleration of the particle.

Top Courses for JEEView all

Physics for JEE Main & Advanced

Mock Tests for JEE Main and Advanced 2025

Chemistry for JEE Main & Advanced

Mathematics (Maths) for JEE Main & Advanced

Chapter-wise Tests for JEE Main & Advanced

Top Courses for JEE

Top Courses for JEE

Suggested Free Tests

Related Content

JEE Courses

Mock Test Series

Past Year Papers

Additional Courses

Company