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 = 3.00 s and (b) at 3.00 s + Dt. (c) Evaluate the limit of Dx/Dt as Dt approaches zero, to find the velocity at

t = 3.00 s.

Answer 1

a)

$x= 3t^{2}$

Position at t =3.00 s

$x= 3(3.00)^{2}=27.00 m$

b)

position at $t =3.00+\Delta t$

$x= 3(3.00+\Delta t)^{2}=3(9.00+(\Delta t)^{2}+6.00\Delta t)$

$x= 27.00+3(\Delta t)^{2}+18.00\Delta t$

(c)

$\frac{\Delta x}{\Delta t}|_{t=3.00s}=\frac{x(3.00+\Delta t)-x(3.00)}{\Delta t}$

$\frac{\Delta x}{\Delta t}|_{t=3.00s}=\frac{27.00+3(\Delta t)^{2}+18.00\Delta t-27.00}{\Delta t}=18.00+3\Delta t$

$\xrightarrow[\Delta t \rightarrow 0]{Lim}\frac{\Delta x}{\Delta t}|_{t=3.00s}=18.00m/s$