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.


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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

	$Here is the method i used .$ $x (t) = (3 m / s^{2}) t^{2}$ $this is the position of the particle at time t .$ $in a later time (t + Δ t) the position of the particle$ $is given by x = 3 {(t + Δ t)}^{2}$ $\Rightarrow x = 3 t^{2} + 6 t Δ t + 3 {(Δ t)}^{2}$ $hence Δ x = x_{f} - x_{i} = 3 t^{2} + 6 t Δ t + 3 {(Δ t)}^{2} - 3 t^{2}$ $= 6 t Δ t + 3 {(Δ t)}^{2}$ $we define velocity v = \lim_{Δ t \to 0} \frac{Δ x}{Δ t} =$ $now \frac{Δ x}{Δ t} = \frac{Δ t (6 t + 3 Δ t)}{Δ t} = 6 t + 3 Δ t$ $now v = \lim_{Δ t \to 0} (6 t + 3 Δ t) = 6 t$
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