Question

The velocity of a particle v at any instant is

$\stackrel{\rightharpoonup }{v}=y\stackrel{\rightharpoonup }{i}+x\stackrel{\rightharpoonup }{j}$

The equation of trajectory of the particle is:

• A. $x^2+y^2=cons\tan t$
• B. $y^2=x^2+cons\tan t$
• C. xy=constant
• D. None of these

Answer 1

The correct option is B

$y^2=x^2+cons\tan t$

Explanation for the correct answer:-

Option (B) $y^2=x^2+cons\tan t$

1. The kinematic equation is used to describe how a body is positioned, as well as its velocity and acceleration in relation to a frame of reference.
2. The rate of change of displacement as velocity.
3. By using the integration approach, velocity may be calculated from the Newton equation.
4. The acceleration equation is produced by integrating velocity.
5. The starting velocity will be 0 if the motion begins at rest and the frame of reference is the same.
6. The frame of reference should be the same if the motion begins at rest.

Equation integration results in the distance equation.
Then the velocity along $x$ axis

$\Rightarrow \frac{dx}{dt}=ky$

Then the velocity along $y$ axis

$\Rightarrow \frac{dy}{dt}=kx$

To find out

$$\begin{gathered} \Rightarrow \frac{dy}{dx}=\frac{{\displaystyle \frac{dy}{dt}}}{{\displaystyle \frac{dx}{dt}}} \\ \Rightarrow \frac{dy}{dx}=\frac{x}{y} \end{gathered}$$

cross multiply the terms we get

$\Rightarrow ydy=xdx$

Now by integration we get

$y^2=x^2+C$

Explanation for the incorrect answer:-

Option (A) $x^2+y^2=cons\tan t$

1. The frame of reference should be the same since the motion begins at rest. assuming a zero starting velocity
2. The acceleration equation is produced by integrating the velocity equation.
3. Option is incorrect.

Option (C) $xy=cons\tan t$

1. The length of an object's route may or may not be equal to its displacement.
2. Speed is the ratio of distance to time.
3. Option is incorrect

Option (D) None of these

1. None of these option is incorrect because the integration of the terms and the coefficients are $y^2=x^2+C$

Therefore, option (B) is the correct answer.