Total Distance Traveled

[Pi Man]

Hello everyone at FMH! This is my first post and hopefully not my last!

Anyway, I got a bit of an embarrassing situation here. We were just assigned the word problem section in our text book and I am a bit lost. I'm an overachiever which is why I've decided to register on a math help site, so do not think of me as another 'college boy BS'n his midterms'.

Anyway, my problem lies in this equation:

f(t) = t^3 - 12t^2 + 36t

Which stands for a particle moving according to the law of motion, where t is measured in seconds and s is in feet.

Also, s = f(t) t >= 0

I've taken the derivative and got the velocity function. No problems so far. I did however stumble across the part of the problem that reads:

"Find the distance traveled during the first 8 feet."

Now I am lost. I have no idea how to calculate that. Please help.

Also, if someone could clearly explain when the particle is moving in a positive direction and why I would much appreciate that too. I got the answer, but I am unsure as to how I got it honestly...

 

[mmm4444bot]

f(t) = t^3 - 12t^2 + 36t

Which stands for a particle moving

"Find the distance traveled during the first 8 seconds."

Your description of the number f(t) as a moving particle is not sufficient to demonstrate that you understand the given scenario.

f(t) is not a particle; it is a number.

Is the particle moving back and forth on some line? Did they explain this in lecture?

Could we think of f(t) as the y-coordinate at time t of some particle moving up and down the y-axis, for example?

What is f(t) exactly? You need to get this, first. After you understand f(t), I think that you'll also understand what the question asks and how to calculate the answer.

Let us know what you think the number f(t) represents, if you need additional guidance. Cheers :cool:

 

[mmm4444bot]

clearly explain when the particle is moving in a positive direction and why

When its velocity is positive, the object moves in the positive direction.

When its velocity is negative, the object moves in the negative direction.

Why, you ask? Do you remember velocity as RISE/RUN, when the movement is in a straight (non-vertical) line on the xy-plane?

When the velocity is negative, either the RISE or the RUN is going down or backwards, respectively. When you go backwards or down, you're not going in the positive direction. :cool:

 

[Pi Man]

Your description of the number f(t) as a moving particle is not good enough.

f(t) is not a particle; it is a number.

Is the particle moving back and forth on some line?

Could we think of f(t) as the y-coordinate at time t of some particle moving up and down the y-axis, for example?

What is f(t) exactly? You need to get this, first. After you understand f(t), I think that you'll also understand what the question asks and how to calculate the answer.

Let us know what you think the number f(t) represents, if you need additional guidance. Cheers :cool:

Ok, this is the problem word for word now. Does this clear it up?

A particle moves according to a law of motion s = f(t), t >= 0, where t is measured in seconds and s in feet.

f(t) = t^3 -12t^2 + 36t

Find the total distance traveled during the first 8 feet.

I do not know how to calculate this.

EDIT - I made a slight mistake the first time I posted this. I said 8 seconds instead of 8 feet. My bad. But I'm still lost.

 

[mmm4444bot]

This is the problem word for word now. Does this clear it up?

Oh, please excuse me. It was not my intent to request clarification of the exercise. I understand the exercise; your first request for help left me thinking that you had no idea what was going on with the given scenario. Sorry. :cool:

So, you understand that the particle is moving back and forth on some line; I think of this line as the positive y-axis, and we see from the exercise that the particle starts at the origin.

Hence, the coordinates of the particle at any time are (0, f(t)). In other words, symbol s stands for the distance of the particle from the origin.

You need to find the points on the y-axis where the particle stops and changes direction (this happens twice during the first eight seconds -- look at a graph of y=f(x)). Then, you simply add up the resulting distances to get the total.

When the particle's velocity is zero, that's when it stops, yes? The first derivative of f(x) is the velocity, yes? In other words, you need to find the values of t that cause the velocity f`(x) to be zero.

Step 1: Determine f`(x)

Step 2: Solve f`(x) = 0

Please show us what you get, up to this point.

Cheers

 

[HallsofIvy]

Ok, this is the problem word for word now. Does this clear it up?

A particle moves according to a law of motion s = f(t), t >= 0, where t is measured in seconds and s in feet.

f(t) = t^3 -12t^2 + 36t

Find the total distance traveled during the first 8 feet.

I do not know how to calculate this.

EDIT - I made a slight mistake the first time I posted this. I said 8 seconds instead of 8 feet. My bad. But I'm still lost.

Your "correction" makes no sense! It will have moved 8 feet in the first 8 feet, of course. Why would they ask such a thing?
The total distance moved in the first 8 seconds makes sense. The total distance moved, by the time the object is 8 feet from the initial point, would be harder but still make sense.

If an object moves 10 feet from the starting point, then 2 feet back towards the starting point, it would be 8 feet from the starting point but will have moved a total of 12 feet. Do you see the point?

 

[Pi Man]

The book gives the answer as 96ft/s

And I'm almost sure that the s is seconds now. The problem reads: "Find the total distance traveled during the first 8 s.

I couldn't tell if it meant feet or seconds.

Can someone CLEARLY explain how it got that answer now?

EDIT: The derivative set equal to 0 yields two values: 2, and 6. So these are the points (2,0) and (6,0) right? Where do you go from there?

 

[KindofSlow]

First, do you understand that the particle is moving in one dimension and what this means?
Second, if yes, do you understand that f(t) is the distance from the starting point?
Third, if yes again, over what periods of time is the particle moving farther away from the starting point and over what periods of time is the particle moving back towards the starting point?

 

[mmm4444bot]

Okay, you now know that the particle stops and reverses direction at 2 seconds, and it does so again at 6 seconds. After 8 seconds, we don't care.

That's three line segments to consider, in the particle's 8-second journey.

Where is the particle, at 2 seconds? In other words, how far did it move in the first 2 seconds? Function f tells you how far: f(2).

Where is the particle at t = 6? Calculate how far it moved during those latest four seconds.

Where is the particle at t = 8? Calculate how far it moved during those latest two seconds.

Add up the distances (three segments of its trip), and you'll have the total distance traveled during the entire 8 seconds.

 

[mmm4444bot]

The book gives the answer as 96 ft/s

Are you sure that your book states 96 feet per second?

It should simply say 96 ft.

The answer has to be a distance, not a velocity.

The derivative set equal to 0 yields two values: 2, and 6. So these are the points (2,0) and (6,0) right?

Yes, but those are two points on the graph of f`(x).

Do not confuse f(x) with f`(x).

In other words, the points (2,0) and (6,0) do not represent the position of the particle. They represent zero velocity at 2 seconds and 6 seconds on the velocity graph (a graph which we do not care about, in this exercise).

The position of the particle at 2 seconds is (0,f(2)) -- if we use my suggestion to think of the particle's path as up-and-down the positive y-axis. The position at 6 seconds is (0,f(6)).

But, you are free to think of this particle traveling on any line you like. If the particle is moving back and forth on the x-axis, then its position at 2 seconds is (f(2),0) and at 6 seconds is (f(6),0).

Really, what's important is determining the distance traveled from one position to the next.

The expression f(2) - f(0) is the difference between the 2-second's position and the starting position. That is the distance traveled during the first 2 seconds, so do the arithmetic to find out how many feet it represents.



Regarding s, you need to consider the context, when you see this symbol in s=f(t) and in phrases like "during the first 8 s".

The symbol s in the assignment s = f(t) is clearly a variable because f(t) is a variable.

The symbol s in the English phrase "during the first 8 s" is clearly an abbreviation for seconds.

It makes no sense to interpret the English phrase "during the first 8 s" as "during the first 8*f(t)", so I do not understand why you're confusing the two.

 

[mmm4444bot]

PS: How long has it been since you finished precalculus? Are you attending high school or college?