# Characteristics of a particles path- v(t) and a(t)

Gold Member

## Homework Statement

The position of a particle moving along a coordinate axis is given by $s(t) = t^3-9t^2+24t+4$ where t is greater than 0.

a) Find v(t)
b) At what time(s) is the particle at rest?
c) On what time intervals is the particle moving from left to right and at what times is it moving from right to left?
d) Sketch the path of the particle

## The Attempt at a Solution

My question is in regards to part d, in sketching the graph of the particle's travel.
To give you some information, I'll post the answers I got to parts a-c.

a) $v(t) = 3t^2-18t+24$
b) The particle is at rest at t=2 second, and at t=4 seconds
c) The particle is moving from left to right on the intervals [0,2) and (4,∞).
The particle is moving from right to left on the interval (2,4)

Now how can I visually represent this? If I use a traditional X,Y coordinate plane, to move backwards it would look like I was going back in time.

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Mark44
Mentor

## Homework Statement

The position of a particle moving along a coordinate axis is given by $s(t) = t^3-9t^2+24t+4$ where t is greater than 0.

a) Find v(t)
b) At what time(s) is the particle at rest?
c) On what time intervals is the particle moving from left to right and at what times is it moving from right to left?
d) Sketch the path of the particle

## The Attempt at a Solution

My question is in regards to part d, in sketching the graph of the particle's travel.
To give you some information, I'll post the answers I got to parts a-c.

a) $v(t) = 3t^2-18t+24$
b) The particle is at rest at t=2 second, and at t=4 seconds
c) The particle is moving from left to right on the intervals [0,2) and (4,∞).
The particle is moving from right to left on the interval (2,4)

Now how can I visually represent this? If I use a traditional X,Y coordinate plane, to move backwards it would look like I was going back in time.
The problem states that the particle is moving along a coordinate axis. I would plot values of s(t) for t = 0, 1, 2, 3, 4, and label each point with the time value. For example, s(0) = 4, so the particle is 4 units to the right of 0, with t = 0. It will necessarily change directions when t = 2 and t = 4. You don't want to use a two-axis coordinate system.

Gold Member
Thanks Mark. I didn't realize that further into the text, it gives the graph.
The math laid out makes sense, but the graph has me confused- maybe because I'm used to only seeing X,Y coordinates.
So on the graph, it's clear that from time=0 seconds to time=2 seconds, the particle is traveling from left to right. Then from t=2 second to t=4 seconds, it's traveling from right to left. Then from t=4 seconds and forward, it's traveling from left to right.
But I don't get the numbers on the number line. From my math, these are the positions where the particle changes direction, and where it starts movement. For example, at t=2 seconds, the position function gives $s(2) = (2)^3-9(2)^2+24(2)+4 = 24$. But how does a position function yield only one value, 24? We need an X and a Y for a position.

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Ray Vickson
Homework Helper
Dearly Missed
Thanks Mark. I didn't realize that further into the text, it gives the graph.
The math laid out makes sense, but the graph has me confused- maybe because I'm used to only seeing X,Y coordinates.
So on the graph, it's clear that from time=0 seconds to time=2 seconds, the particle is traveling from left to right. Then from t=2 second to t=4 seconds, it's traveling from right to left. Then from t=4 seconds and forward, it's traveling from left to right.
But I don't get the numbers on the number line. From my math, these are the positions where the particle changes direction, and where it starts movement. For example, at t=2 seconds, the position function gives $s(2) = (2)^3-9(2)^2+24(2)+4 = 24$. But how does a position function yield only one value, 24? We need an X and a Y for a position.
The problem did say "motion along a coordinate axis", so the motion is one-dimensional. For example, if you choose to have the motion going along the $x$-axis, the position at $t=2$ is $(24,0).$

Gold Member
Ahh ok. That makes sense. Thank you guys.