Question

the velocity v of a particle moving along x axis varies with its position (x)=alpha squareroot x where alpha is a constant.Which of the following graph represents the variation of its acceleration(a) with time(t)?

Answer 1

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3) will be the required answer.
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Answer 2

Answer:

The slope is gradient.

3 is correct option.

Explanation:

Given that,

Velocity $v=\alpha\sqrt{x}$

Velocity :

The velocity is the first derivative of the position of the particle.

$v = \dfrac{dx}{dt}$

Acceleration :

The acceleration is the first derivative of the velocity of the particle.

$a = \dfrac{dv}{dt}$

The velocity is

$v=\alpha\sqrt{x}$

$v^2= \alpha^2\times x$

$x = \dfrac{v^2}{\alpha^2}$

On differentiating both side

$\dfrac{dx}{dt}=\dfrac{1}{\alpha^2}\times2v\dfrac{dv}{dt}$

$v=\dfrac{1}{\alpha^2}\times2v\dfrac{dv}{dt}$

$\dfrac{\alpha^2}{2}=a$

Here, $\alpha$ is a constant.

So. The acceleration is constant.

Hence, The slope is gradient.