A-005-QA

course PHY 231

7/10 00:29

If your solution to stated problem does not match the given solution, you should self-critique per instructions at

http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/

file3_259.htm

Your solution, attempt at solution. If you are unable to attempt a solution,

give a phrase-by-phrase interpretation of the problem along with a statement of

what you do or do not understand about it. This response should be given, based

on the work you did in completing the assignment, before you look at the given

solution.

At the end of this document, after the qa problems (which provide you with

questions and solutions), there is a series of Questions, Problems and

Exercises.

005. Uniformly Accelerated Motion

Preliminary notes:

On any interval there are seven essential quantities in terms of which we

analyze the motion of a nonrotating object:

the time interval `dt between the beginning and the end of the interval

the displacement `ds of the object during the interval

the initial velocity v0, the velocity at the beginning of the interval

the final velocity vf, the velocity at the end of the interval

the average velocity vAve of the object during the interval

the change `dv in the velocity of the object during the interval

the average acceleration a_Ave of the object during the interval

You should remember these symbols and their meanings. You will be using them

repeatedly, and you will soon get used to them.

You should at any time be able to list these seven quantities and explain the

meaning of each.

In any question or problem that involves motion, you should identify the

interval of interest, think about what each of these quantities means for the

object, and identify which quantities can be directly determined from the given

information.

You will of course improve your understanding and appreciation of these

quantities as you work through the qa and the associated questions and problems.

Note also that `dt = t_f - t_0, where t_f represents the final clock time and

t_0 the initial clock time on the interval, and that `ds = s_f - s_0, where s_f

represents the final position and t_0 the initial position of the object on the

interval.

Further discussion of symbols (you can just scan this for the moment, then refer

to it when and if you later run into confusion with notation)

the symbol x is often used instead of s for the position of an object moving

along a straight line, so that `dx might be used instead of `ds, where `dx = x_f

- x_0

some authors use either s or x, rather that s_f or x_f, for the quantity that

would represent final position on the interval; in particular the quantity we

express as `dx might be represented by x - x_0, rather than x_f - x_0

some authors use t instead of `dt; there are good reasons for doing so but at

this point in the course it is important to distinguish between clock time t and

time interval `dt; this distinction tends to be lost if we allow t to represent

a time interval

the quantity we refer to as `dt is often referred to as 'elapsed time', to

distinguish it from 'clock time'; once more we choose here to use different

symbols to avoid confusion at this critical point in the course)

If the acceleration of an object is uniform, then the following statements

apply. These are important statements. You will need to answer a number of

questions and solve a number of problems in order to 'internalize' their

meanings and their important. Until you do, you should always have them handy

for reference. It is recommended that you write a brief version of each

statement in your notebook for easy reference:

1. A graph of velocity vs. clock time forms a straight line, either level or

increasing at a constant rate or decreasing at a constant rate.

2. The average velocity of the object over any time interval is equal to the

average of its velocity at the beginning of the time interval (called its

initial velocity) and its velocity at the end of the time interval (called its

final velocity).

3. The velocity of the object changes at a constant rate (this third statement

being obvious since the rate at which the velocity changes is the acceleration,

which is assumed here to be constant).

4. The acceleration of the object at every instant is equal to the average

acceleration of the object.

*********************************************

Question: `q001. Note that there are 9 questions in this assignment.

Suppose that an object increases its velocity at a uniform rate, from an initial

velocity of 5 m/s to a final velocity of 25 m/s during a time interval of 4

seconds.

By how much does the velocity of the object change?

What is the average acceleration of the object?

What is the average velocity of the object?

(keep your notes on this problem, which is continued through next few questions)

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

'dv = 25m/s - 5m/s = 20m/s

a_avg = (20m/s) / 4s = 5m/s^2

v_avg = (25m/s + 5m/s)0.5 = 15m/s

confidence rating #$&* 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

The velocity of the object changes from 5 meters/second to 25 meters/second so

the change in velocity is 20 meters/second. The average acceleration is

therefore (20 meters/second) / (4 seconds) = 5 m / s^2. The average velocity of

the object is the average of its initial and final velocities, as asserted

above, and is therefore equal to (5 meters/second + 25 meters/second) / 2 = 15

meters/second (note that two numbers are averaged by adding them and dividing by

2).

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

Self-critique rating #$&*

*********************************************

Question: `q002. How far does the object of the preceding problem travel in

the 4 seconds?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

'ds = v_avg*'dt = 4s*15m/s = 60m

confidence rating #$&* 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

The displacement `ds of the object is the product vAve `dt of its average

velocity and the time interval, so this object travels 15 m/s * 4 s = 60 meters

during the 4-second time interval.

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

Self-critique rating #$&*

*********************************************

Question: `q003. Explain in commonsense terms how we determine the

acceleration and distance traveled if we know the initial velocity v0, and final

velocity vf and the time interval `dt.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

by having initial and final velocity you can find the average by adding the two

and dividing by two. To find the change in velocity subtract v_i from v_f. From

these we can determing the acceleration (as change in velocity over change in

time) as well as determine distance travelled from the average velocity being

multiplied by te change in time.

confidence rating #$&* 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

In commonsense terms, we find the change in velocity since we know the initial

and final velocities, and we know the time interval, so we can easily calculate

the acceleration. Again since we know initial and final velocities we can

easily calculate the average velocity, and since we know the time interval we

can now determine the distance traveled.

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

Self-critique rating #$&*

*********************************************

Question: `q004. Symbolize the situation by first giving the expression for

the acceleration in terms of v0, vf and `dt, then by giving the expression for

vAve in terms of v0 and vf, and finally by giving the expression for the

displacement in terms of v0, vf and `dt.

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

'dv = v_f - v_i = 25m/s - 5m/s = 20m/s

a_avg = 'dv / 'dt = (20m/s) / 4s = 5m/s^2

v_avg = (v_i + v_f)0.5 = (25m/s + 5m/s)0.5 = 15m/s

'ds = v_avg*'dt

Therefore,

a_avg = 'dv/'dt = (v_f - v_i) / 'dt

v_avg = (v_i + v_f)0.5

'ds = (v_i + v_f)0.5*'dt

confidence rating #$&* 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

The acceleration is equal to the change in velocity divided by the time

interval; since the change in velocity is vf - v0 we see that the acceleration

is a = ( vf - v0 ) / `dt.

The average velocity is the average of the initial and final velocities, which

is expressed as (vf + v0) / 2.

When this average velocity is multiplied by `dt we get the displacement, which

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

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary): OK

Self-critique rating #$&*

*********************************************

Question: `q006. This situation is identical to the previous, and the

conditions implied by uniformly accelerated motion are repeated here for your

review: If the acceleration of an object is uniform, then the following

statements apply:

1. A graph of velocity vs. clock time forms a straight line, either level or

increasing at a constant rate or decreasing at a constant rate.

2. The average velocity of the object over any time interval is equal to the

average of its velocity at the beginning of the time interval (called its

initial velocity) and its velocity at the end of the time interval (called its

final velocity).

3. The velocity of the object changes at a constant rate (this third statement

being obvious since the rate at which the velocity changes is the acceleration,

which is assumed here to be constant).

4. The acceleration of the object at every instant is equal to the average

acceleration of the object.

Describe a graph of velocity vs. clock time, assuming that the initial velocity

occurs at clock time t = 0.

At what clock time is the final velocity then attained?

What are the coordinates of the point on the graph corresponding to the initial

velocity (hint: the t coordinate is 0, as specified here; what is the v

coordinate at this clock time? i.e., what is the velocity when t = 0?).

What are the coordinates of the point corresponding to the final velocity?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

a) it starts at (0,5) and moves in a straight line in a direction of slope 5

b) final velocity is attained at 4 seconds

c) 5m/sec^2

d) (4, 25)

confidence rating #$&* 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

The initial velocity of 5 m/s occurs at t = 0 s so the corresponding graph point

is (0 s, 5 m/s). The final velocity of 25 meters/second occurs after a time

interval of `dt = 4 seconds; since the time interval began at t = 0 sec it ends

at at t = 4 seconds and the corresponding graph point is ( 4 s, 25 m/s).

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

Self-critique rating #$&*

*********************************************

Question: `q007. Is the v vs. t graph increasing, decreasing or level between

the two points, and if increasing or decreasing is the increase or decrease at a

constant, increasing or decreasing rate?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

Increasing between the two points at a constant rate.

confidence rating #$&* 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

Since the acceleration is uniform, the graph is a straight line. The graph

therefore increases at a constant rate from the point (0, 5 m/s) to the point (4

s, 25 m/s).

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

Self-critique rating #$&*

*********************************************

Question: `q008. What is the slope of the graph between the two given points,

and what is the meaning of this slope?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

5m/sec^2

confidence rating #$&* 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

The rise of the graph is from 5 m/s to 25 m/s and is therefore 20 meters/second,

which represents the change in the velocity of the object. The run of the graph

is from 0 seconds to 4 seconds, and is therefore 4 seconds, which represents the

time interval during which the velocity changes. The slope of the graph is rise

/ run = ( 20 m/s ) / (4 s) = 5 m/s^2, which represents the change `dv in the

velocity divided by the change `dt in the clock time and therefore represents

the acceleration of the object.

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

Self-critique rating #$&*

*********************************************

Question: `q009. The graph forms a trapezoid, starting from the point (0,0),

rising to the point (0,5 m/s), then sloping upward to (4 s, 25 m/s), then

descending to the point (4 s, 0) and returning to the origin (0,0). This

trapezoid has two altitudes, 5 m/s on the left and 25 m/s on the right, and a

base which represents a width of 4 seconds. What is the average altitude of the

trapezoid and what does it represent, and what is the area of the trapezoid and

what does it represent?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

The average altitude is the average velocity. It is 15m/sec

confidence rating #$&*

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

The two altitudes are 5 meters/second and 25 meters/second, and their average is

15 meters/second. This represents the average velocity of the object on the

time interval. The area of the trapezoid is equal to the product of the average

altitude and the base, which is 15 m/s * 4 s = 60 meters. This represents the

product of the average velocity and the time interval, which is the displacement

during the time interval.

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

Self-critique rating #$&*

You should submit the above questions, along with your answers and self-

critiques. You may then begin work on the Questions, Problem and Exercises, as

instructed below.

Questions, Problems and Exercises

You should answer the questions and work the problems as given below, as

appropriate to your course. Your work should normally be handwritten, should

include diagrams and sketches where appropriate, and should go into your Physics

notebook (not into your lab notebook).

If the course is not specified for a problem, then students in all physics

courses should do that problem.

Principles of Physics students need not do the questions or problems that are

specified for General College Physics or University Physics.

General College Physics students need not do questions or problems specified for

University Physics.

University Physics students should do all questions and problems.

Principles of Physics students may if they wish do some of the questions and

problems specified for General College Physics, but this is neither expected nor

required. These problems are accessible to Principles of Physics students, but

are generally more challenging that what is expected of students in this course.

(Some University Physics problems will also be accessible to Principles of

Physics students, though some will not.)

General College Physics students who wish to do so are welcome to work some or

all of the University Physics questions and problems, though this is neither

expected nor required. Many of the University Physics questions and problems

are more challenging than those expected of General College Physics students,

and a number of the problems require the use of calculus, which is not expected

of General College Physics students.

You are not expected to submit these questions and problems. It would take too

much of your time to key in all the answers and solutions. The Query at the end

of the assignment will ask you selected questions, which you will at that time

be expected to answer based on the work you have done in your notebook.

Questions related to q_a_

1. If we know the initial velocity v0, the final velocity vf and the time

interval `dt for the motion of an object, then

How do we find its average acceleration for the interval?

How do we find its displacement during the interval?

How does each of your answers follow from the definition an average rate of

change?

2. Suppose we have two points on a straight-line graph of velocity vs. clock

time.

How do we construct a trapezoid to represent the motion on the intervening

interval?

What aspect of the graph represents the change in velocity for the interval, and

why?

What aspect of the graph represents the change in clock time for the interval,

and why?

What aspect of the graph represents the acceleration for the interval, and why?

What aspect of the graph represents the displacement for the given interval, and

why?

3. If we know v0, `dt and a for motion on an interval:

What quantity can we determine using the definition of acceleration?

What quantity can we then determine using the newly determined quantity with one

of the given quantities?

At this point, what two quantities remain to be determined?

Which quantity can be determined using two of the five quantities we now know?

How can we use the six quantities we now know to find the value of the seventh?

Questions related to Introductory Problem Sets

1. In the figure below the given quantities v0, vf and `dt are listed in a

straight line across the middle of the diagram.

There are lines from v0 and vf to vAve, and also from v0 and vf to `dv. Explain

how v0 and vf are related to vAve, and also how they are related to `dv.

There are 'red' lines from vAve and `dt to `ds. How are vAve and `dt related to

`ds by the definition of average velocity?

There are 'green' lines from `dv and `dt to a. How are `dv and `dt related to a

by the definition of average acceleration?

Questions related to Class Notes

1. If a cart is found to have acceleration 30 cm/s^2 when it rolls freely down

an incline having slope .05, and acceleration 70 cm/s^2 on an incline having

slope .08, then at what average rate is its acceleration changing with respect

to ramp slope?

Questions/problems for Principles of Physics Students

4. About how long would it take an automobile moving at the speed limit to

travel from Washington, D.C. to Cleveland, Ohio? You may base your answer on

any reasonable estimate of the distance and the speed limit, but you should

specify your estimates and show how you get from your estimates to your answer.

5. Estimate how long it would take a runner to run from New York to California.

You may base your answer on any reasonable estimate of the distance and the

running speed, but you should specify your estimates and show how you get from

your estimates to your answer.

Questions/problems for General College Physics Students

6. Estimate how many liters of water a human being drinks in a lifetime. Would

this water fill an average swimming pool?

Questions/problems for University Physics Students

What is the angle between the between the vectors -2i+6j and 2i - 3j ?

What is the cross product A X B of vector A, whose magnitude is 12.0 meters and

which is directed along the negative y axis, and the vector B, with magnitude

18.0 m and directed at 37 deg as measured counterclockwise from the positive x

axis? Give both the magnitude and the direction of the cross product. "

&#Your work looks very good. Let me know if you have any questions. &#

#$&*