A particle moves along positive branch of the curve y=x / 2;...

A particle moves along positive branch of the curve $y=x / 2$; where $x=t^{3} / 3, x$ and $y$ are measured in metres and $t$ in seconds, then (a) the velocity of particle at $t=1 \mathrm{~s}$ is $\hat{i}+(1 / 2) \hat{j}$ (b) the velocity of particle at $t=1 \mathrm{~s}$ is $(1 / 2) \hat{i} \mid \hat{j}$ (c) the acecleration of particle at $t=2 \mathrm{~s}$ is $2 \hat{i}+\hat{j}$ (d) the acceleration of particle at $t=2 \mathrm{~s}$ is $\hat{i}$ ? $2 \hat{j}$

A particle moves along positive branch of the curve $y=x / 2$; where $x=t^{3} / 3, x$ and $y$ are measured in metres and $t$ in seconds, then
(a) the velocity of particle at $t=1 \mathrm{~s}$ is $\hat{i}+(1 / 2) \hat{j}$
(b) the velocity of particle at $t=1 \mathrm{~s}$ is $(1 / 2) \hat{i} \mid \hat{j}$
(c) the acecleration of particle at $t=2 \mathrm{~s}$ is $2 \hat{i}+\hat{j}$
(d) the acceleration of particle at $t=2 \mathrm{~s}$ is $\hat{i}$ ? $2 \hat{j}$

Key Concepts

Parametric Equations for Particle Motion

In kinematics, particle motion is often described using parametric equations where each coordinate (such as x and y) is written as a function of an independent parameter, usually time. This approach allows tracing the particle’s path in space and calculating its dynamics by differentiating the position functions with respect to time.

Velocity as the Derivative of Position

Velocity is defined as the rate of change of the position vector with respect to time. In the context of parametric motion, this involves differentiating each component of the position vector separately to obtain the velocity components, which can then be combined into a vector using standard unit vectors.

Acceleration as the Derivative of Velocity

Acceleration represents the rate of change of the velocity vector with respect to time. By differentiating the velocity components with respect to time, one derives the acceleration components. This process is central to understanding the dynamics of the particle as it provides insight into how the particle’s speed and direction evolve.

Chain Rule in Differentiation

The chain rule is a fundamental tool in calculus used to differentiate composite functions. When dealing with parametric equations, it often becomes necessary to apply the chain rule to relate the rates of change between different variables, especially when one variable is defined implicitly as a function of another.

Vector Representation and Component Analysis

Working with vectors is essential for analyzing motion in two or more dimensions. This involves breaking down vectors, like velocity and acceleration, into their individual components along mutually perpendicular axes. Component analysis simplifies computation and provides clear insight into how each axis contributes independently to the overall motion.

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