A force acts on a 3 gram particle such that its position x=3...

A force acts on a 3 gram particle such that its position $x=3 t-4 t^{2}+t^{3}$, where $x$ is in metre and $t$ is in second. The work done during first $4 \mathrm{~s}$ is
(A) $825 \mathrm{~mJ}$
(B) $285 \mathrm{~mJ}$
(C) $528 \mathrm{~mJ}$
(D) Zero

Key Concepts

Unit Conversion

Accurate problem solving in physics requires consistency in units. This often involves converting units such as mass (e.g., grams to kilograms) or energy (e.g., joules to millijoules) to ensure that all quantities are correctly measured and applied in formulas.

Calculus in Kinematics

Differentiation and integration are fundamental in kinematics. Differentiation is used to derive velocity and acceleration from a position-time function, while integration is used to calculate quantities like work when force varies with time. These techniques are essential for analyzing and understanding motion and energy transfer.

Work-Energy Theorem

This theorem asserts that the work done by the net force acting on a particle is equal to the change in its kinetic energy. By calculating the particle’s velocity from its position function and then determining its kinetic energy at different times, one can obtain the work done on the particle over a given time interval.

Newton’s Second Law

This principle connects force, mass, and acceleration, stating that the force acting on a particle is equal to its mass multiplied by its acceleration. When position is expressed as a function of time, acceleration can be determined by taking the second derivative of the position function, which is essential for calculating the force acting on the particle.

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