Question

A particle is moving with speed v=b√x along positive x-axis. Calculate the speed of the particle at time t=T

Answer 1

Given that,

Speed $v=b\sqrt{x}$

Time $t=\tau$

We know that,

Velocity :

The velocity of the particle is equal to the rate of change of position of the particle.

In mathematically,

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

Acceleration :

The acceleration of the particle is equal to the rate of change of velocity of the particle.

In mathematically,

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

We need to calculate the speed of the particle at time

Using formula of velocity

$v=b\sqrt{x}$

On differentiating

$\dfrac{dv}{dt}=\dfrac{b}{2\sqrt{x}}\dfrac{dx}{dt}$

$a=\dfrac{b}{2\sqrt{x}}\times v$

Put the value into the formula

$a=\dfrac{b}{2\sqrt{x}}\times\dfrac{b}{\sqrt{x}}$

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

$\dfrac{dv}{dt}=\dfrac{b^2}{2}$

On integrating

$\int_{0}^{v}{\dfrac{dv}{dt}}=\int_{0}^{\tau}{\dfrac{b^2}{2}}$

$\int_{0}^{v}dv=\int_{0}^{\tau}{\dfrac{b^2}{2}}dt$

$v=(\dfrac{b^2}{2}t)_{0}^{\tau}$

$v=\dfrac{b^2}{2}\tau$

Hence, The speed of the particle at time is $\dfrac{b^2}{2}\tau$

Answer 2

Explanation:

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