A particle moves along the positive branch of the curve y=(x^2)/(2) where x=(t^2)/(2), where x a

step 1 We will learn about some basic concept of the kinematics. The question is, a particle moves alon

A particle moves along the positive branch of the curve...

A particle moves along the positive branch of the curve $y=\frac{x^{2}}{2}$ where $x=\frac{t^{2}}{2}$, where $x$ and $y$ are measured in metre and $t$ in second. At $t=2 \sec$, the velocity of the particle is: (a) $(2 \hat{1}-4 \hat{j}) \mathrm{m} / \mathrm{sec}$ (b) $\left.\left(2 \hat{1}+4^{2}\right)\right) \mathrm{m} / \mathrm{sec}$ (c) $(2 \hat{1}+2 \hat{j}) \mathrm{m} / \mathrm{sec}$ (d) $(4 \hat{1}-2 \hat{j}) \mathrm{m} / \mathrm{sec}$

A particle moves along the positive branch of the curve $y=\frac{x^{2}}{2}$ where $x=\frac{t^{2}}{2}$, where $x$ and $y$ are measured in metre and $t$ in second. At $t=2 \sec$, the velocity of the particle is:
(a) $(2 \hat{1}-4 \hat{j}) \mathrm{m} / \mathrm{sec}$
(b) $\left.\left(2 \hat{1}+4^{2}\right)\right) \mathrm{m} / \mathrm{sec}$
(c) $(2 \hat{1}+2 \hat{j}) \mathrm{m} / \mathrm{sec}$
(d) $(4 \hat{1}-2 \hat{j}) \mathrm{m} / \mathrm{sec}$

Key Concepts

Differentiation in Kinematics

Differentiation is the tool used to obtain velocity and acceleration from the position functions in kinematics. By differentiating the position function with respect to time, one finds the rate of change of position, which is the velocity. This process is fundamental in solving problems involving motion, where determining instantaneous rates of change is required.

Velocity Vector Components

The velocity vector of a moving particle consists of its components in the coordinate directions. It is obtained by differentiating the position components with respect to time. Understanding the vector nature of velocity allows for the analysis of both the magnitude and the direction of the motion. This decomposition is central when dealing with motion along curved paths.

Parametric Equations

Parametric equations describe the position of a particle in terms of an independent parameter, typically time. In kinematics, they are useful because they allow the motion in each coordinate direction to be expressed as separate functions of time. Understanding how to work with these equations is essential for analyzing the trajectory and instantaneous properties of motion, such as velocity and acceleration.

Chain Rule in Differentiation

The chain rule is a fundamental principle in calculus that is used when differentiating composite functions. When a variable depends on another which in turn depends on a third, such as y being a function of x, and x being a function of t, the derivative of y with respect to t is obtained by multiplying the derivative of y with respect to x by the derivative of x with respect to t. This is crucial when relating the motion along a curve defined by a relation between x and y.

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