Question 123: Motion in 2D & Vectors

Question

A particle is moving with a velocity
$$\overrightarrow v \, = K(y\widehat i + x\widehat j),$$ where K is a constant. 
The general equation for its path is :

Accepted Answer

Given,
$$\overrightarrow v  = K\left( {y\widehat i + x\widehat j} ight)$$
$$ 	herefore $$   Velocity in x direction,
$${v_x} = {{dx} \over {dt}} = Ky\,$$   . . . . . (1)
Velocity in y direction,
v
y
= $$\,{{dy} \over {dt}}$$ = Kx . . . . . . . (2)
$$ 	herefore $$   $${{\,{{dy} \over {dt}}} \over {{{dx} \over {dt}}}} = {{Kx} \over {Ky}}$$
$$ \Rightarrow $$   $${{dy} \over {dx}} = {x \over y}$$
$$ \Rightarrow $$   ydy $$=$$ xdx
Integrating both sides we get,
$$\int {ydx}  = \int {xdx} $$
$$ \Rightarrow $$   $${{{y^2}} \over 2} = {{{x^2}} \over 2} + c$$
$$ \Rightarrow $$   $${y^2} = {x^2} + 2c$$
$$ 	herefore $$   General equation,
y
2
= x
2
+ constant.

Answer

y = x2 + constant

Answer

y2 = x + constant

Answer

y2 = x2 + constant

Answer

xy = constant

Motion in 2D & Vectors

A particle is moving with a velocity $\overrightarrow v \, = K(y\widehat i + x\widehat j),$ where K is a constant. The general equation for its path is :

Choose the correct answer: