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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 = γt, where γ is a positive constant of appropriate dimensions. Which of the following statements is (are) true?

The force applied on the particle is constant

The force is conservative

The distance of the particle from the origin increases linearly with time

The speed of the particle is proportional to time

Solution:

dk/dt = γt and k = 1/2mv²
Differentiating k = 1/2mv² w.r.t. time
∴dk/dt = mvdv/dt = γt
∴m∫₀ᵛ vdv = γ∫₀ᵗ tdt
mv²/2 = γt²/2
v = √(γ/m) t → v ∝ t; So option (D) is correct
Since F = ma
∴F = m√(γ/m) = √(γm) = constant
So option (A) is correct
Since force applied is constant, it is conservative in nature, So option (B) is correct
Option (A), (B) and (D) are correct answers.