asst_7_query

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ݯ|ڠō͔assignment #007

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007. `query 7

Physics I

02-05-2008

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18:42:13

Describe the flow diagram you would use for the uniform acceleration situation in which you are given v0, vf, and `dt.

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RESPONSE -->

From v0 and vf we get vAve (v0 +vf)/2 and `dv (vf-v0) on the first level. Then from vAve and `dt we get `ds on the second level. We derive a from `dv/`dt.

confidence assessment: 2

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18:42:30

** We start with v0, vf and `dt on the first line of the diagram.

We use vO and vf to find Vave, indicated by lines from v0 and vf to vAve.

Use Vave and 'dt to find 'ds, indicated by lines from vAve and `dt to `ds.

Then use `dv and 'dt to find acceleration, indicated by lines from vAve and `dt to a. **

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RESPONSE -->

self critique assessment: 3

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18:45:45

Describe the flow diagram you would use for the uniform acceleration situation in which you are given `dt, a, v0

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RESPONSE -->

We would get vf using vf = v0 + a `dt on the first level. Then we can get vAve from vf and v0 on the second level using (v0 + vf)/2. On the third level we get `ds using vAve * `dt.

confidence assessment: 2

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18:48:09

** Student Solution: Using 'dt and a, find 'dv.

Using 'dv and v0, find vf.

Using vf and vO, find vave.

Using 'dt and Vave, find 'ds. **

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RESPONSE -->

I did this in a somewhat different order, but I think the fundamentals are the same.

self critique assessment: 2

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21:50:05

Explain in detail how the flow diagram for the situation in which v0, vf and `dt are known gives us the two most fundamental equations of motion.

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RESPONSE -->

vAve is derived from (vf - v0)/2 and this value is multiplied by time to derive `ds. so the equation:

`ds = vAve * `dt

becomes:

`ds = (vf - v0)/2 * `dt

This is the second fundamental equation of uniform motion.

The first fundamental equation is:

vf = v0 + a`dt

a`dt represents `dv, so plugging in that value, we are able to calculate vf given a knowledge of v0.

confidence assessment: 1

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21:51:22

**Student Solution:

v0 and vf give you `dv = vf - v0 and vAve = (vf + v0) / 2.

`dv is divided by `dt to give accel. So we have a = (vf - v0) / `dt.

Rearranging this we have a `dt = vf - v0, which rearranges again to give vf = v0 + a `dt.

This is the second equation of motion.

vAve is multiplied by `dt to give `ds. So we have `ds = (vf + v0) / 2 * `dt.

This is the first equation of motion

Acceleration is found by dividing the change in velocity by the change in time. v0 is the starting velocity, if it is from rest it is 0. Change in time is the ending beginning time subtracted by the ending time. **

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RESPONSE -->

I think my reasoning was pretty close to the example, and I feel pretty confident that I understand the first two fundamental equations, but I wasn't feeling confident that I answered the question properly.

self critique assessment: 2

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21:59:22

qaExplain in detail how the flow diagram for the situation in which v0, a and `dt are known gives us the third fundamental equations of motion.

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RESPONSE -->

If we have v0, a, and `dt the we can compute vf using the first fundamental equation and figure out everything else in the intuitive way. However if we take the first fundamental equation and substitue the right hand side of it:

v0 +a`dt

into the second fundamental equation:

`ds = (v0 +a`dt + v0) /2 * `dt

then we can simplify to the third fundamental equation:

`ds = (v0 +a`dt + v0) /2 * `dt

`ds = (v0 + .5a`dt) `dt

`ds = `dt v0 + .5a dt^2

self critique assessment: 2

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22:01:15

** a and `dt give you `dv. `dv and v0 give you vf. v0 and vf give you vAve. vAve and `dt give you `ds.

In symbols, `dv = a `dt.

Then vf = v0 + `dv = v0 + a `dt.

Then vAve = (vf + v0)/2 = (v0 + (v0 + a `dt)) / 2) = v0 + 1/2 a `dt.

Then `ds = vAve * `dt = [ v0 `dt + 1/2 a `dt ] * `dt = v0 `dt + 1/2 a `dt^2. **

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RESPONSE -->

I think I undertand most of this. I can follow the examples that you give, but I have a hard time synthesizing them on my own when it comes to the 3rd and 4th fundamental equations.

self critique assessment: 2

To understand the third equation:

If an object moves at constant velocity v0 for time interval `dt then it moves through displacement v0 `dt.

If an object accelerates from rest at acceleration a for time `dt then it attains velocity a `dt; averaging this with initial velocity 0 gives average velocity 1/2 a `dt. Multiplying this by time interval `dt gives us displacement vAve * `dt = 1/2 a `dt * `dt = 1/2 a `dt^2.

So the equation

`ds = v0 `dt + 1/2 a `dt^2

says that the displacement is equal to the displacement v0 `dt that would occur at the initial velocity, plus the additional displacement 1/2 a `dt^2 due to the acceleration.

Another way to see it:

Initial velocity is v0. Change in velocity is a `dt. So final velocity is v0 + a `dt.

Average velocity is average of initial and final velocities, (v0 + (v0 + a `dt) ) / 2 = (2 v0 + a `dt) / 2 = v0 + 1/2 a `dt.

Displacement is therefore `ds = vAve * `dt = (v0 + 1/2 a `dt) * `dt = v0 `dt + 1/2 a `dt^2.

To understand the fourth equation:

vf^2 = v0^2 + 2 a `ds so that

vf^2 - v0^2 = 2 a `ds.

So the product 2 a `ds is the change in the squared velocity.

Compare this with the fact that a `dt is the change in the velocity.

If you put acceleration together with `dt you get information about the change in velocity.

If you put acceleration together with `ds you get information about the change in the squared velocity.

Later we will see that this describes the distinction between momentum and energy.

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22:02:43

Why do we think in terms of seven fundamental quantities while we model uniformly accelerated motion in terms of five?

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RESPONSE -->

Using vAve and `dv makes it easier to intuitively understand situations of uniform motion, even though it is generally better to model things by using the fewest possible variables.

confidence assessment: 2

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22:03:00

** ONE WAY OF PUTTING IT:

The four equations are expressed in terms of five fundamental quantities, v0, vf, a, `dt and `ds. However to think in terms of meanings we have to be able to think not only in terms of these quantities but also in terms of average velocity vAve and change in velocity `dv, which aren't among these five quantities. Without the ideas of average velocity and change in velocity we might be able to use the equations and get some correct answers but we'll never understand motion.

ANOTHER WAY:

The four equations of unif accelerated motion are expressed in terms of five fundamental quantities, v0, vf, a, `dt and `ds.

The idea here is that to intuitively understand uniformly accelerated motion, we must often think in terms of average velocity vAve and change in velocity `dv as well as the five quantities involved in the four fundamental equations.

one important point is that we can use the five quantities without any real conceptual understanding; to reason things out rather than plugging just numbers into equations we need the concepts of average velocity and change in velocity, which also help us make sense of the equations. **

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RESPONSE -->

self critique assessment: 3

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22:04:45

Accelerating down an incline through a given distance vs. accelerating for a given time

Why does a given change in initial velocity result in the same change in final velocity when we accelerated down a constant incline for the same time, but not when we accelerated down the same incline for a constant distance?

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RESPONSE -->

If we use distance as the measure then initial velocity causes us to cover that given distance at a different rate. If we use time as the measure then our `dv variable will be the same no matter what our initial velocity.

confidence assessment: 2

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22:04:56

** If we accelerate down a constant incline our rate of change of velocity is the same whatever our initial velocity.

So the change in velocity is determined only by how long we spend coasting on the incline. Greater `dt, greater `dv.

If you travel the same distance but start with a greater speed there is less time for the acceleration to have its effect and therefore the change in velocity will be less.

You might also think back to that introductory problem set about the car on the incline and the lamppost. Greater initial velocity results in greater average velocity and hence less time on the incline, which gives less time for the car to accelerate. **

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RESPONSE -->

self critique assessment: 3

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