A particle of mass m moving in one dimension whose...

A particle of mass $m$ moving in one dimension whose displacement is $x$ under the action of a force, isrelated to time $t$ (in sec) by $t=\sqrt[3]{x}+6$. The displacement of the particle when its velocity is zero, will be: (a) $2 \mathrm{~m}$ (b) $4 \mathrm{~m}$ (c) $0 \mathrm{~m}$ (d) $6 \mathrm{~m}$

A particle of mass $m$ moving in one dimension whose displacement is $x$ under the action of a force, isrelated to time $t$ (in sec) by $t=\sqrt[3]{x}+6$. The
displacement of the particle when its velocity is zero, will be:
(a) $2 \mathrm{~m}$
(b) $4 \mathrm{~m}$
(c) $0 \mathrm{~m}$
(d) $6 \mathrm{~m}$

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Key Concepts

Inverse Function Transformation

Occasionally, functions are given in a form where time is expressed as a function of displacement, requiring the inversion of the relationship to express displacement in terms of time. This transformation is crucial for applying differentiation techniques effectively and for understanding the direct relationship between position and time in the context of motion.

Displacement and Velocity Relationship in Kinematics

In kinematics, the displacement function describes the position of a particle as a function of time, while velocity is defined as the derivative of the displacement with respect to time. Understanding this relationship is fundamental to analyzing motion, as the velocity indicates how fast and in which direction the position is changing at any given moment.

Differentiation in Motion Analysis

Differentiation is a core calculus technique used to determine the rate at which a quantity changes. When applied to a displacement function, differentiation yields the velocity function, which can then be analyzed to find specific characteristics such as when the velocity is zero. Standard rules of differentiation, including the power and chain rules, are often used in this process.

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