SATHEE: Kinematics 6 Question 5

Kinematics 6 Question 5

5. A particle is moving with a velocity $v = k (y \hat{i} + x \hat{j})$ , where $k$ is a constant.

The general equation for its path is

(2019 Main, 9 Jan I)

(a) $y = x^{2} +$ constant

(b) $y^{2} = x +$ constant

(d) $y^{2} = x^{2} +$ constant

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Answer:

Correct Answer: 5. (d)

Solution:

Given, velocity of a particle is

$v = k (y \hat{i} + x \hat{j})$

Suppose, it’s position is given as

$\begin{aligned} r & = x \hat{i} + y \hat{j} \\ ∴ & v & = \frac{d \underset{―}{r}}{d t} & = \frac{d}{d t} (x \hat{i} + y \hat{j}) = \frac{d x}{d t} \hat{i} + \frac{d y}{d t} \hat{j} \end{aligned}$

Comparing Eqs. (i) and (ii), we get

and

$\begin{aligned} \frac{d x}{d t} = y \\ \frac{d y}{d t} = x \end{aligned}$

Dividing Eq. (iii) and Eq. (iv), we get

$\frac{\frac{d x}{d t}}{\frac{d y}{d t}} = \frac{y}{x} \Rightarrow x \frac{d x}{d t} = y \frac{d y}{d t} or x d x = y d y$

Integrating both sides, we get

$\int x d x = \int y d y or \frac{x^{2}}{2} + \frac{c_{1}}{2} = \frac{y^{2}}{2} + \frac{c_{2}}{2}$

where, $c_{1}$ and $c_{2}$ are the constants of integration.

$\begin{aligned} \Rightarrow x^{2} + c = y^{2} [here, c (constant) = c_{1} - c_{2}] \\ or y^{2} = x^{2} + constant \end{aligned}$

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Kinematics 6 Question 5

5. A particle is moving with a velocity $v = k (y \hat{i} + x \hat{j})$ , where $k$ is a constant.

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

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अन्य संसाधन

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सदस्यता

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Kinematics 6 Question 5

5. A particle is moving with a velocity v=k(yi^+xj^), where k is a constant.

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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5. A particle is moving with a velocity $v = k (y \hat{i} + x \hat{j})$ , where $k$ is a constant.