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Question

The position of a particle moving along the x axis varies in time according to the expression x=3t2, where x is in meters and t is in seconds. Evaluate its position (a) at t=3.00s and (b) at 3.00s+Δt. (c) Evaluate the limit of Δx/Δt as Δt approaches zero to find the velocity at t=3.00s.

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Solution

a) At any time, t, the position is given by x=(3.00m/s2)t2.
Thus, at ti=3.00s:xi=(3.00m/s2)(3.00s)2=27.0m.

(b) At tf=3.00s+Δt::xf=(3.00m/s2)(3.00s+Δt)2, or
xf=27.0m+(18.0m/s)Δt+(3.00m/s2)(Δt)2.

(c) The instantaneous velocity at t=3.00s is:
limΔt0ΔxΔt=limΔt0(18.0m/s)Δt+(3.00m/s2)(Δt)2Δt
=limΔt0(18.0m/s)+(3.00m/s2)(Δt)=18.0m/s

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