#### Experiment: Uniformity of acceleration on a constant slope

#### The equations of uniformly accelerated motion

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

*We attempt to determine by an experiment whether the acceleration of
a friction cart on a constant incline depends on its velocity or its position on an
incline. We then derive the four equations of uniformly accelerated motion.
Finally we consider the difference between the effect on final velocity of a given
acceleration through a distance, with different initial velocities, with its effect over a
time interval with different initial velocities.*

Having defined acceleration, we attempt to establish at least **one set of
conditions **under which **acceleration should be uniform**, namely a
car coasting down a **constant incline**. We take data for later
analysis.

We then formulate the **consequences **of uniform acceleration, using

- the definition of average velocity
**vAve = `ds / `dt**and the fact that for uniform acceleration**vAve = (vf + v0) / 2** - the definition of average acceleration
**aAve = (vf - v0) / `dt**, which for constant acceleration tells us the**vf = v0 + d `dt** - algebraic
**elimination of `dt**from the two equations - algebraic
**elimination of `vf**from the two equations.

We finally see that from uniform **acceleration and time interval**, we
can determine **change in velocity **while from uniform **acceleration
and distance **we determine the **change in the squared velocity**.

The following questions arise:

- Why do we say that the first equation of uniformly accelerated motion expresses the
**definition of average velocity**, while the second expresses the**definition of acceleration**? - Why, for
**uniform acceleration**, is vAve = (vf + v0) / 2, while this is**not usually true for nonuniform acceleration**? - In commonsense terms, why does
**change of velocity**over a**given distance**, with a**given uniform acceleration**, differ with**initial velocity**?

We begin with an experiment to test whether on an inclined plane with a **constant
slope**, the **acceleration seems to be constant **regardless of **velocity
**or **position **on the slope.

- We time a low-friction dynamics
card as it
**accelerates from rest for different distances**down a steady incline. - If our calculations of
**average acceleration**for the**various distances**all give us**nearly the same result**, with**only random fluctuations**in rather than a steady pattern to the unavoidable differences, then we will have**strong evidence**that**acceleration**is in fact**determined only by the slope**.

We begin with the return to the situation in which we know `ds, a and v0. Recall that
the **two most fundamental equations of motion **are

vf = v0 + a `dt,

which was derived by considering what could be concluded from knowledge of v0, a and `dt, and which indicates the by-now obvious fact that when we add the change in velocity `dv = a `dt to the initial velocity we get the final velocity, and

`ds = (vf + v0) / 2 * `dt,

derived from the initially known quantities v0, vf and `dt, which indicates that to get the displacement we multiply the average velocity by the time interval.

Would have also seen that knowing v0, a and `dt, are natural line of reasoning leads to
the **third fundamental equation**

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

This equation could have been obtained from the first two by substituting the right-hand side of the first equation for vf in the second and doing the straightforward algebraic simplification.

We note that the two most fundamental equations have the three variables vf, v0 and `dt in common. We've just observed that by eliminating vf we get the third fundamental equation of motion. We can also see that this equation doesn't help in the present situation, where we want to draw some kind of a conclusion from knowledge of `ds, a and v0. None of the three equations we have so far contains more than two of these variables among its four variables, and hence none can be solved for one of the remaining variables.

If we **eliminate `dt from the first two equations **we can see the we
will end up with an equation involving vf, v0, a and `ds, which contains three of the
known variables and will therefore be useful in the present situation.

The **equation **that we get when we **eliminate `dt **is **`ds
= ( vf^2 - v0^2 ) / (2a)**, which we rearrange into the form

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

This is the **fourth fundamental equation of motion**.

We observe that we now have **four
equations**, each with **four variables**, with the property that **given
any three **of the variables v0, vf, `ds, `dt and a, at least one of these
equations will **contain the three **and can hence be **solved for the
fourth**.

These equations all arise from the first two; these two we understand at the intuitive level as expressing the definition of acceleration and average velocity.

The details of the process of eliminating `dt from the two most fundamental equations are shown below.

- The primary law of algebra that we have used is the intuitively obvious distributive law of multiplication over addition.
- We haven't used any formulas like (a+b)(a-b) = a^2 - b^2 (or worse, the pedagogical atrocity FOIL, which obscures the distributive law and applies only to multiplication of two binomials, leaving the problem of multiplying trinomials and longer expressions untouched), which students might or might not recall.
- The final step, where we see that ds = (vf^2 - v0^2) / (2a), is a simple exercise in using the distributive law and is left to the reader.

We now pose the following question:

If a cart coasts a **specific distance **down a certain incline **starting
from rest**, it will experience a certain **velocity change**.

- If it coasts the
**same distance**down the same incline,**starting at a higher velocity**, and if we accept that the**acceleration is identical**in both cases, would we not expect the**same velocity change?**

Most students will agree that **this argument seems persuasive**, and that
since the **conditions seee to be the same**, the **velocity change **will
indeed be the **same **in both cases.

**However**, it could be argued that the cart starting with**higher velocity**will spend**less time**on the incline, and hence will spend**less time accelerating**, with the result that it**picks up less speed**.

Before you read further you should carefully **consider both of these arguments **and
decide for yourself **which seems to be more persuasive**.

The **equations of motion **settle the argument.

- The first argument says essentially that since a and `ds are identical, the velocity difference should be the same, while
- the second says that since a and `dt are different velocity difference should be the same.

The equation **vf = v0 + a `dt **tells us clearly that if we have the **same
acceleration **over a **shorter time interval**, our **velocity
will increase by less**. The change in velocity is without a doubt **a `dt**.

The equation **vf^2 = v0^2 + 2 a `ds**, on the other hand, tells us that
when **a and `ds are identical**, the difference in the **squared
velocities **will be identical:

- If
**a and `ds**are identical, then**2 a `ds**will be identical. - Since
**vf^2 - v0^2 = 2 a `ds**, it follows that**vf^2 - v0^2**will be**the same**for both cases.

Solving for a `dt and a `ds, we obtain the equations

a `dt = vf - v0

a `ds = (vf^2 - v0^2) / 2,

which will both have profound implications later when we study **momentum and
energy**.

- It will turn out that what we call
**momentum**is associated with the**velocity**v of an object, while what we define as energy is associated with**v^2**. - Thus
**a `dt**will be associated with**change in momentum**, while**a `ds**is associated with**change in energy**.

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