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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.
$$\mathrm{The}\:\mathrm{position}\:\:\mathrm{as}\:\mathrm{a}\:\mathrm{function}\:\mathrm{of}\:\mathrm{time}\:{x}\left({t}\right)\:\mathrm{for}\:\mathrm{a}\:\mathrm{particle}\:\mathrm{in} \\ $$$$\mathrm{motion}\:\mathrm{is}\:\mathrm{given}\:\mathrm{as}\:\:{x}\left({t}\right)\:=\:\left(\mathrm{3}\:\mathrm{m}/\mathrm{s}^{\mathrm{2}} \right){t}^{\mathrm{2}} \:.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{velocity} \\ $$$$\mathrm{of}\:\mathrm{this}\:\mathrm{particle}\:\mathrm{as}\:\mathrm{a}\:\mathrm{function}\:\mathrm{of}\:\mathrm{time}. \\ $$
Answered by Dwaipayan Shikari last updated on 11/Aug/20
x(t)=3(m/s^2 ).t^2 ((dx(t))/dt)=6t v(t)=(6(m/s).)t
$${x}\left({t}\right)=\mathrm{3}\frac{{m}}{{s}^{\mathrm{2}} }.{t}^{\mathrm{2}} \\ $$$$\frac{{dx}\left({t}\right)}{{dt}}=\mathrm{6}{t} \\ $$$${v}\left({t}\right)=\left(\mathrm{6}\frac{{m}}{{s}}.\right){t} \\ $$
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 + Δt) the position of the particle is given by x = 3(t + Δt)^2 ⇒ x = 3t^2 + 6tΔt +3(Δt)^2 hence Δx = x_f −x_i = 3t^2 + 6tΔt + 3(Δt)^2 −3t^2 = 6tΔt + 3(Δt)^2 we define velocity v = lim_(Δt→0) ((Δx)/(Δt)) = now ((Δx)/(Δt)) = ((Δt(6t + 3Δt))/(Δt)) = 6t + 3Δt now v = lim_(Δt→0) (6t + 3Δt) = 6t
$$\mathrm{Here}\:\mathrm{is}\:\mathrm{the}\:\mathrm{method}\:\mathrm{i}\:\mathrm{used}. \\ $$$$\:{x}\left({t}\right)\:=\:\left(\mathrm{3}\:\mathrm{m}/\mathrm{s}^{\mathrm{2}} \right){t}^{\mathrm{2}} \\ $$$$\mathrm{this}\:\mathrm{is}\:\mathrm{the}\:\mathrm{position}\:\mathrm{of}\:\mathrm{the}\:\mathrm{particle}\:\mathrm{at}\:\mathrm{time}\:{t}.\: \\ $$$$\mathrm{in}\:\mathrm{a}\:\mathrm{later}\:\mathrm{time}\:\left({t}\:+\:\Delta{t}\right)\:\mathrm{the}\:\mathrm{position}\:\mathrm{of}\:\mathrm{the}\:\mathrm{particle} \\ $$$$\mathrm{is}\:\mathrm{given}\:\mathrm{by}\:{x}\:=\:\mathrm{3}\left({t}\:+\:\Delta{t}\right)^{\mathrm{2}} \: \\ $$$$\Rightarrow\:{x}\:=\:\mathrm{3}{t}^{\mathrm{2}} \:+\:\mathrm{6}{t}\Delta{t}\:+\mathrm{3}\left(\Delta{t}\right)^{\mathrm{2}} \\ $$$$\mathrm{hence}\:\Delta{x}\:=\:{x}_{{f}} \:−{x}_{{i}} \:=\:\mathrm{3}{t}^{\mathrm{2}} \:+\:\mathrm{6}{t}\Delta{t}\:+\:\mathrm{3}\left(\Delta{t}\right)^{\mathrm{2}} −\mathrm{3}{t}^{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{6}{t}\Delta{t}\:+\:\mathrm{3}\left(\Delta{t}\right)^{\mathrm{2}} \\ $$$$\mathrm{we}\:\mathrm{define}\:\mathrm{velocity}\:{v}\:=\:\underset{\Delta{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\Delta{x}}{\Delta{t}}\:=\: \\ $$$$\mathrm{now}\:\frac{\Delta{x}}{\Delta{t}}\:=\:\frac{\Delta{t}\left(\mathrm{6}{t}\:+\:\mathrm{3}\Delta{t}\right)}{\Delta{t}}\:=\:\mathrm{6}{t}\:+\:\mathrm{3}\Delta{t} \\ $$$$\mathrm{now}\:{v}\:=\:\underset{\Delta{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\mathrm{6}{t}\:+\:\mathrm{3}\Delta{t}\right)\:=\:\mathrm{6}{t}\: \\ $$

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