Question

Each of four particles move along an x axis. Their coordinates (in meters) as functions of time
(in seconds) are given by
particle 1: x(t) = 3.5 − 2.7t3
particle 2: x(t) = 3.5 +2.7t3
particle 3: x(t) = 3.5 +2.7t2
particle 4: x(t) = 3.5 − 3.4t − 2.7t2
Which of these particles is speeding up for t > 0?
A. All four
B. Only 1
C. Only 2 and 3
D. Only 2, 3, and 4
E. None of them
ans: A

Answer 1

$$\begin{gathered} \mathit{x}\mathbf{}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{}\mathbf{=}\mathbf{}\mathbf{3}\mathbf{.}\mathbf{5}\mathbf{}\mathbf{-}\mathbf{}\mathbf{2}\mathbf{.}\mathbf{7}\mathbf{}{\mathit{t}}^{\mathbf{3}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{\Rightarrow }\mathbf{}\mathbf{3}\mathbf{.}\mathbf{5}\mathbf{}\mathbf{-}\mathbf{}\mathbf{2}\mathbf{.}\mathbf{7}\mathbf{}{\mathit{t}}^{\mathbf{3}}\mathbf{}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathit{t}\mathbf{}\mathbf{=}\sqrt[\mathbf{3}]{\mathbf{}\frac{\mathbf{3}\mathbf{.}\mathbf{5}}{\mathbf{2}\mathbf{.}\mathbf{7}}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\mathbf{.}\mathbf{13}\mathbf{}\mathbf{}\mathit{s}\mathit{e}\mathit{c} \end{gathered}$$
for 0 < t < 1.13 the particle is speeding up in posive x - axis and for t > 1.13 S speeding up in negative x - axis .
for the equations x ( t) = 3.5 + 2.7 t³ and
x ( t ) = 3.5 + 2.7 t²​ the particle
is speeding up in positive x - axis .
hence option (C ) is corret .