Question

A particle is moving with velocity $\vec v$ = K(y $\widehat {i}$ + x $\widehat {j}$) , where \[{\text{K}}\] is a constant. The general equation for its path is:
A. \[{{\text{y}}^{\text{2}}} = {{\text{x}}^{\text{2}}} + {\text{constant}}\]
B. \[{\text{y}} = {{\text{x}}^{\text{2}}} + {\text{constant}}\]
C. \[{{\text{y}}^{\text{2}}} = {\text{x}} + {\text{constant}}\]
D. \[{\text{yx}} = {\text{constant}}\]

Answer 1

Hint: First of all, we will find out the component of the velocity along the horizontal and the vertical component separately. Then we will find out the relation between \[{\text{x}}\] and \[{\text{y}}\] using calculus. After that we will integrate the relation and obtain the result.

Complete step by step answer:
Given,
$\vec v$ = K(y $\widehat {i}$ + x $\widehat {j}$)
So, we get,
\[{{\text{v}}_{\text{x}}} = {\text{Ky}}\] , this is because while taking the velocity in horizontal direction, the vertical component is absent.
So,
\[\dfrac{{{\text{dx}}}}{{{\text{dt}}}} = {\text{Ky}}\]
Again, we have,
\[{{\text{v}}_{\text{y}}} = {\text{Kx}}\] , this is because while taking the velocity in vertical direction, the horizontal component is absent.
So,
\[\dfrac{{{\text{dy}}}}{{{\text{dt}}}} = {\text{Kx}}\]
Then we divide both equations and we get,
$\implies$ $\dfrac{{{\text{dy}}}}{{{\text{dx}}}} = \dfrac{{{\text{dy}}}}{{{\text{dt}}}} \div \dfrac{{{\text{dx}}}}{{{\text{dt}}}} = \dfrac{{{\text{Kx}}}}{{{\text{Ky}}}} = \dfrac{{\text{x}}}{{\text{y}}} \\$
$\implies$ $ {\text{ydy}} = {\text{xdx}} \\$
Integrating the equation, we get,
\[\int {{\text{ydy}}} = \int {{\text{xdx}}} \]
$\therefore$ \[\dfrac{{{{\text{y}}^{\text{2}}}}}{{\text{2}}} = \dfrac{{{{\text{x}}^{\text{2}}}}}{{\text{2}}} + {\text{c}}\]

Hence, the required answer is \[{{\text{y}}^{\text{2}}} = {{\text{x}}^{\text{2}}} + {\text{c}}\], option A.

Additional Information:
Velocity: The velocity of an object is the rate at which its location is changed relative to a reference frame and depends on the time. The velocity is the defined speed and direction of movement of an object. The principle of velocity, a branch of classical mechanics that describes the movement of bodies, is central in kinematics. Speed is a physical quantity of the vector; it is necessary for defining both magnitude and direction. The scalar absolute value (magnitude) of velocity is called speed, a consistent derived unit calculating the quantity as meters per second ( \[{\text{m}}{{\text{s}}^{ - 1}}\] ) in the SI system. "\[5\,{\text{m}}{{\text{s}}^{ - 1}}\]" is, for instance, a scalar; "\[5\,{\text{m}}{{\text{s}}^{ - 1}}\] east" is a vector. If the speed, direction and both are changed, the object has a change in speed and acceleration is said to occur.

Note:
It is important to note that $\widehat {i}$ and $\widehat {j}$ represents unit vector along the horizontal and the vertical direction, which has unit magnitude and the direction along the desired axis. A point is lying on the horizontal axis if $\widehat {j}$ is zero whereas the point is lying on the vertical axis if $\widehat {i}$ is zero.