0910 Quiz

1.  What is the definition of acceleration?

Definition:  Rate of change of velocity with respect to clock time.

then from definition of rate:  change in velocity / change in clock time, also expressed as `dv / `dt and identical to the slope of a position vs. clock time graph between two points.

if we know velocity and clock time at two clock times:

vAve = (vf - v0) / (tf - t0)

2.  List the four equations of uniformly accelerated motion.

`ds = (vf + v0) / 2 * `dt, from definition of average velocity and linearity of v vs. t graph

vf = v0 + a `dt, from definition of acceleration and linearity of v vs. t graph

`ds = v0 `dt + .5 a `dt^2, eliminating vf from first two

vf^2 = v0^2 + 2 a `ds, eliminating `dt from first two

3.  List at least six of questions you ask when you wish to interpret a graph.

What do the extremes mean?

What do slopes mean?

What are the extremes and zeros of the slopes and what does that tell you?

What do areas mean?

4.  What does the slope of a v vs. t graph tell you and how is this connected to the definitions?

rise is change in v

run is change in t

slope = rise / run = change in v / change in t = average rate of change of v with respect to t or average acceleration

5.  What does the area beneath a v vs. t graph tell you and how is this connected to the definitions?

average altitude is approximate average v

width is `dt

area is approx. vAve * `dt.

Since vAve = `ds / `dt area is approx. `ds / `dt * `dt = `ds, which is displacement or change in position.

6.  If you have a graph of position vs. clock time, how do you construct a graph of velocity vs. clock time?

Look for clock times where the slope is 0; at these clock times velocity will be 0 and the point will be on the horizontal axis of the v vs. t graph.

Look for clock times where slope is greatest or least.  At these clock times the velocity will be greatest or least.

Look for the intervals where the slope is positive/negative.  On these intervals velocity will be positive/negative.

Look for the intervals where the slope is increasing/decreasing.  On these intervals velocity will be increasing/decreasing.

For a series of t values find the slopes of the position vs. clock time graph

Graph slope vs. t.

7.  If you have a graph of velocity vs. clock time how do you construct a graph of position vs. clock time?

divide the region under the curve into a series of approximating trapezoids

find the areas

the area of each trapezoid is the change in the position corresponding to that time interval

add the position change to the previous position to get the new position

8.  List the seven quantities in terms of which we understand uniformly accelerated motion.

v0, vf, vAve, `dv, `ds, `dt, a

9.  If you were given v0, vf and `ds for a uniform-acceleration situation, how would you reason out vAve, `dv, `dt and a without using equations?  Include a flow diagram as demonstrated in the Introductory Problem Sets.

List v0, vf and `ds at the top.

From v0 and vf we can find vAve, so draw a couple of lines from v0 and vf and join then at a node called vAve.

From v0 and vf we can find `v=dv, so draw a couple of lines from v0 and vf and join then at a node called `dv.

From vAve and `ds we can find `dt, since vAve = `ds / `dt, so we draw lines from vAve and `ds and join then at a node called `dt.

Finally from `dv and `dt we can find a = `dv / `dt, so we draw lines from `dv and `dt and join then at a node called a.

10.  What would be your strategy for using the equations of uniform acceleration to answer the question on #9?

The given information includes v0, vf and `ds.

We can solve the first equation `ds = (v0 + vf) / 2 * `dt for `dt, obtaining `dt = 2 `ds / (vf + v0).

We can then solve the second equation vf = v0 + a `dt for a, obtaining a = (vf - v0) / `dt (which shouldn't surprise us, since the second equation is just a rearrangement of this definition of average acceleration).

Using the second equation with the given info and the `dt we got from the first will give us the correct solution if we didn't mess up when we found `dt.

If we use the fourth equation vf^2 = v0^2 + 2 a `ds to find a, we get a = (vf^2 - v0^2) / (2 `ds), which gives us a in terms of our given information, not relying on anything we might have messed up.  The result should be the same as what we got for a in the preceding.

Integral and derivative terminology

The derivative is the slope of the graph.

For example the derivative of the velocity vs. clock time graph at a given clock time is the slope of the v vs. t graph at that clock time, which as everyone knows represents the acceleration at that clock time.

A definite integral is the area under the graph between two points.

For example the definite integral of the v vs. t graph between t = 3 and t = 7 is the area under the graph between t = 3 and t = 7, which as we know corresponds to the displacement between t = 3 and t = 7.

Experiment:

Make the observations necessary to determine the acceleration of a steel ball rolling down an incline vs. the percent incline.

Sketch a graph of acceleration vs. percent incline and find its slope.