How Do You Analyze Particle Motion Along the X-axis Using Calculus?

[noname09]

A particle moves along the x-axis according to the equation s(t) = 1/3t^3 -t^2 -8t +12, where s is the directed distance (in meters) of the particle from the origin at time t (in seconds). Find
a. the directed distance of the particle from the origin, its velocity and acceleration at the following time instants: t = 0, 1, 2, 3, 4, 5, 6. Describe the particle’s motion at these time instants.
b. the time instant/s when the particle is instantaneously at rest.
c. the time instant/s when the particle is in uniform motion.
d. the time interval/s when the particle is moving to the right/left.
e. the time interval/s when the particle is accelerating/decelerating.
f.

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The trace of the particle’s motion on the x-axis

Hello, I'm having a hard time setting up and solving this problem. any help would be appreciated. Thank you!

 

[MarkFL]

Hello and welcome to MHB! (Wave)

We are given:

\(\displaystyle s(t)=\frac{1}{3}t^3-t^2 -8t +12\)

What this tells us is that at time $t$, the particle is located at $s(t)$. Now for velocity $v$ and acceleration $a$ we may use the following definitions:

\(\displaystyle v(t)\equiv\d{s}{t}\)

\(\displaystyle a(t)\equiv\d{v}{t}\)

Using these definitions, can you explicitly state the velocity and acceleration of the particle as functions of $t$?