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

In summary: 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).##Ahh ok. That makes sense. Thank you guys.
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opus
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Homework Statement


Please see the attached images.

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

Homework Equations

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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  • #2
opus said:

Homework Statement


Please see the attached images.

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

Homework Equations

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.
 
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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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  • #4
opus said:
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).##
 
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  • #5
Ahh ok. That makes sense. Thank you guys.
 

FAQ: Characteristics of a particles path- v(t) and a(t)

What is the relationship between velocity and time for a particle's path?

The velocity of a particle's path is represented by the slope of its position-time graph. This means that the steeper the slope, the greater the velocity, and vice versa.

How does acceleration affect a particle's path?

Acceleration is the rate of change of velocity, and it can affect a particle's path in several ways. If the acceleration is positive, the particle's velocity will increase, causing it to move faster along its path. If the acceleration is negative, the particle's velocity will decrease, causing it to slow down along its path. Additionally, if the acceleration is perpendicular to the velocity, it can cause the particle to change direction.

Can a particle's path have a constant velocity and changing acceleration?

Yes, this is possible. In this case, the particle's path would be a straight line, but the slope of its position-time graph would be changing, indicating a changing acceleration. This can happen when the particle is experiencing a constant force, such as gravity, while moving in a straight line.

How do you calculate the average velocity and acceleration of a particle's path?

The average velocity of a particle's path can be calculated by dividing the change in position by the change in time. Similarly, the average acceleration can be calculated by dividing the change in velocity by the change in time. These calculations can also be represented graphically by finding the slope of the position-time and velocity-time graphs, respectively.

What factors can affect the shape of a particle's path?

A particle's path can be affected by various factors, such as the initial velocity and position, the magnitude and direction of the applied force, and the presence of other forces or objects in the environment. The shape of the path can also be influenced by the type of motion, such as linear, circular, or projectile motion.

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