The-position-as-a-function-of-time-x-t-for-a-particle-in-motion-is-given-as-x-t-3-m-s-2-t-2-Find-the-velocity-of-this-particle-as-a-function-of-time-

Question Number 107533 by Rio Michael last updated on 11/Aug/20

The position as a function of time x (t) for a particle in

motion is given as x (t) = (3 m / s^{2}) t^{2} . Find the velocity

of this particle as a function of time .

Answered by Dwaipayan Shikari last updated on 11/Aug/20

x (t) = 3 \frac{m}{s^{2}} . t^{2}

\frac{d x (t)}{d t} = 6 t

v (t) = (6 \frac{m}{s} .) t

Answered by Rio Michael last updated on 11/Aug/20

Here is the method i used .

x (t) = (3 m / s^{2}) t^{2}

this is the position of the particle at time t .

in a later time (t + Δ t) the position of the particle

is given by x = 3 {(t + Δ t)}^{2}

\Rightarrow x = 3 t^{2} + 6 t Δ t + 3 {(Δ t)}^{2}

hence Δ x = x_{f} - x_{i} = 3 t^{2} + 6 t Δ t + 3 {(Δ t)}^{2} - 3 t^{2}

= 6 t Δ t + 3 {(Δ t)}^{2}

we define velocity v = \lim_{Δ t \to 0} \frac{Δ x}{Δ t} =

now \frac{Δ x}{Δ t} = \frac{Δ t (6 t + 3 Δ t)}{Δ t} = 6 t + 3 Δ t

now v = \lim_{Δ t \to 0} (6 t + 3 Δ t) = 6 t