Question

A particle is moving with speed v = √x along positive
X-axis. Calculate the speed of the particle at time t = τ
(assume that the particle is at origin at t = 0)

Answer 1

Given,

$\sf{\longrightarrow v=\sqrt x}$

At $\sf{x=0,\ v=0.}$

Squaring both sides,

$\sf{\longrightarrow x=v^2}$

Differentiating wrt time $\sf{t,}$

$\sf{\longrightarrow\dfrac{dx}{dt}=\dfrac{d}{dt}\left(v^2\right)}$

$\sf{\longrightarrow\dfrac{dx}{dt}=2v\cdot\dfrac{dv}{dt}}$

But,

• $\sf{\dfrac{dx}{dt}=v}$

• $\sf{\dfrac{dv}{dt}=a}$

Then,

$\sf{\longrightarrow v=2va}$

$\sf{\longrightarrow a=\dfrac{1}{2}}$

$\sf{\longrightarrow \dfrac{dv}{dt}=\dfrac{1}{2}}$

$\sf{\longrightarrow dv=\dfrac{1}{2}\ dt}$

Integrating,

• the velocity from 0 to v(τ) where v(τ) is the speed of the particle at time t = τ.

• the time from 0 to τ in order to find the speed at time t = τ.

$\displaystyle\sf{\longrightarrow\int\limits_0^{v(\tau)}dv=\int\limits_0^{\tau}\dfrac{1}{2}\ dt}$

$\displaystyle\sf{\longrightarrow v(\tau)-0=\dfrac{1}{2}(\tau-0)}$

$\displaystyle\sf{\longrightarrow\underline{\underline{v(\tau)=\dfrac{\tau}{2}}}}$