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.
004. Acceleration
Goals:
Question: `q001 Note that there are 13 questions in this assignment.
At a certain instant the speedometer of a car reads 5 meters / second (of course car speedometers in this country generally read speeds in miles per hour and km per hour, not meters / sec; but they could easily have their faces re-painted to read in meters/second, and we assume that this speedometer has been similarly altered). Exactly 4 seconds later the speedometer reads 25 meters/second (that, incidentally, indicates very good acceleration, as you will understand later). At what average rate is the speed of the car changing with respect to clock time?
Your solution:
Confidence rating:
Given Solution:
The rate of change of the speed with respect clock time is equal to the change in the speed divided by the change in the clock time. So we must ask, what is the change in the speed, what is the change in the clock time and what therefore is the rate at which the speed is changing with respect to clock time?
The change in speed from 5 meters/second to 25 meters/second is 20 meters/second. This occurs in a time interval lasting 4 seconds. The average rate of change of the speed is therefore (20 meters/second)/(4 seconds) = 5 meters / second / second. This means that on the average, per second, the speed changes by 5 meters/second.
STUDENT QUESTION
Would we not have s^2 since we are
multiplying s by s?
INSTRUCTOR RESPONSE
That is correct. However in this question I've chosen not to confuse the issue by simplifying the complex fraction m/s/s, which we address separately.
To clarify, m / s / s means, by the
order of operations, (m/s) / s, which is (m/s) * (1/s) = m/s^2.
Self-critique (if necessary):
Self-critique rating:
Question: `q002. Explain in commonsense terms of the significance for an automobile of the rate at which its velocity changes. Do you think that a car with a more powerful engine would be capable of a greater rate of velocity change?
Your solution:
Confidence rating:
Given Solution:
A car whose velocity changes more rapidly will attain a given speed in a shorter time, and will be able to 'pull away from' a car which is capable of only a lesser rate of change in velocity. A more powerful engine, all other factors (e.g., weight and gearing) being equal, would be capable of a greater change in velocity in a given time interval.
STUDENT COMMENT:
The significance for an automobile
of the rate at which its velocity changes is the amount of speed it takes to
travel to a place in a certain amount of time. If one car is traveling along
side another, and they are going to the same location the velocity will be how
long it take this car to get to this location going at a speed other than the
other car. If a car with a more powerful engine were to travel the same distance
its velocity would be capable of a greater rate if increased speed occurred.
INSTRUCTOR RESPONSE:
It's necessary here to distinguish
between velocity, which is a pretty intuitive concept, and rate of change of
velocity, which is much less intuitive and less familiar.
An object can change velocity at a constant rate, from rest to a very high
velocity. All the while the rate of change of velocity with respect to clock
time can be unchanging.
So the rate of change of velocity with respect to clock time has nothing to do
with how fast the object is moving, but rather with how quickly the velocity is
changing.
Moving from one location to another, the displacement is the change in position.
If the displacement is divided by the time required we get the average rate of
change of position with respect to clock time, or average velocity.
Self-critique (if necessary):
Self-critique rating:
Question: `q003. Explain how we obtain the units meters / second / second in our calculation of the rate of change of the car's speed.
Your solution:
Confidence rating:
Given Solution:
When we divide the change in velocity, expressed in meters/second, by the duration of the time interval in seconds, we get units of (meters / second) / second, often written meters / second / second.
Self-critique (if necessary):
Self-critique rating:
Question: `q004. The unit (meters / second) / second is actually a complex fraction, having a numerator which is itself a fraction. Such a fraction can be simplified by multiplying the numerator by the reciprocal of the denominator. We thus get (meters / second) * (1/ second). What do we get when we multiply these two fractions?
Your solution:
Confidence rating:
Given Solution:
Multiplying the numerators we get meters * 1; multiplying the denominators we get second * second, which is second^2. Our result is therefore meters * 1 / second^2, or just meters / second^2. If appropriate you may at this point comment on your understanding of the units of the rate of change of velocity.
Self-critique (if necessary):
Self-critique rating:
Question: `q005. If the velocity of an object changes from 10 m/s to -5 m/s during a time interval of 5 seconds, then at what average rate is the velocity changing with respect to clock time?
Your solution:
Confidence rating:
Given Solution:
We see that the velocity changes from 10 meters/second to -5 meters/second, a change of -15 meters / second, during a five-second time interval. A change of -15 m/s during a 5 s time interval implies an average rate of -15 m/s / (5 s) = -3 (m/s)/ s = -3 m/s^2. This is the same as (-3 m/s) / s, as we saw above. So the velocity is changing by -3 m/s every second.
STUDENT QUESTION
Do you have to do the step -3 m/s /s. Because I get the same answer not doing that.
INSTRUCTOR RESPONSE
Your solution read ' -5 m/s - 10 m/s = -15 m/s / 5 seconds = -3 m/s '.
Everything was right except the units on
your answer. So the answer to you question is 'Yes. It is very
important to do that step.'
The final answer in the given solution is '-3 m/s every second', which is not at
all the same as saying just '-3 m/s'.
-15 m/s / (5 s) = -3 m/s^2, which means -3 m/s per s or -3 cm/s every second or
-3 m/s/s.
-3m/s is a velocity. The question didn't ask for a velocity, but for an average rate of change of velocity.
-3 m/s per second, or -3 m/s every second, or -3 m/s/s, or -3 m/s^2 (all the same) is a rate of change of velocity with respect to clock time.
Self-critique (if necessary):
Self-critique rating:
Question: `q006. You should have noted that velocity can be positive or negative, as can the change in velocity or the rate at which velocity changes. The average rate at which a quantity changes with respect to time over a given time interval is equal to the change in the quantity divided by the duration of the time interval. In this case we are calculating the average rate at which the velocity changes. If v represents velocity then we we use `dv to represent the change in velocity and `dt to represent the duration of the time interval. What expression do we therefore use to express the average rate at which the velocity changes?
Your solution:
Confidence rating:
Given Solution:
The average rate would be expressed by [ave rate of velocity change with respect to clock time] = `dv / `dt. The expression [ave rate of velocity change with respect to clock time] is pretty cumbersome so we give it a name. The name we give it is 'average acceleration', abbreviated by the letter aAve. Using a to represent acceleration, write down the definition of average acceleration. The definition of average acceleration is aAve = `dv / `dt. Please make any comments you feel appropriate about your understanding of the process so far.
STUDENT COMMENT:
It’s average velocity so it would be
aAve.
INSTRUCTOR RESPONSE:
Good, but note:
It’s average acceleration (not average velocity) so it would be aAve.
In most of your course acceleration is constant, so initial accel = final accel
= aAve.
In this case we can just use 'a' for the acceleration.
STUDENT QUESTION
If I understand this correctly,
the average rate in which velocity changes is acceleration???? Where did average
acceleration fit into the problem, the problem asked for the average rate that
velocity changed?
INSTRUCTOR RESPONSE
Acceleration is rate of change of velocity with respect to clock time. So the terms 'average acceleration' and 'average rate of change of velocity with respect to clock time' are identical. The term 'average rate of change of velocity' actually leaves off the 'with respect to clock time', but in the context of uniformly accelerated motion 'average rate of change of velocity' is understood to mean 'average rate of change of velocity with respect to clock time' .
Self-critique (if necessary):
Self-critique rating:
Question: `q007. If a runner is moving at 6 meters / sec at clock time t = 1.5 sec after starting a race, and at 9 meters / sec at clock time t = 3.5 sec after starting, then what is the average acceleration of the runner between these two clock times?
If you can, answer the question as posed. If not, first consider the two questions below:
Your solution:
Confidence rating:
Given Solution:
We see that the runner's velocity changes from 6 meters/second to 9 meters/second, a change of `dv = 9 m/s - 6 m/s = 3 m/s, during a time interval lasting from t = 1.5 sec to t = 3.5 sec.
The duration of the interval is `dt = 3.5 s - 1.5 s = 2.0 s.
The rate at which the velocity changes is therefore 3 m/s / (2.0 s) = 1.5 m/s^2.
STUDENT QUESTION
I'm not understanding why you have the
power of 2 for.
INSTRUCTOR RESPONSE
When you divide m/s by s you do the algebra of the fractions and get m/s^2. You don't get m/s.
The distinction is essential:
m/s^2 is a unit of acceleration.
m/s is a unit of velocity.
Velocity and acceleration are two completely different aspects of motion.
The algebra of dividing m/s by s was given in a previous question in this document. In a nutshell, (m/s) / s = (m/s) * (1/s) = m/s^2.
Self-critique (if necessary):
Self-critique rating:
Question: `q008. On a graph of velocity vs. clock time, we can represent the two events of this problem by two points on the graph. The first point will be (1.5 sec, 6 meters/second) and the second point will be (3.5 sec, 9 meters / sec). What is the run between these points and what does it represent?
What is the rise between these points what does it represent?
What does the slope between these points what does it represent?
Your solution:
Confidence rating:
Given Solution:
The rise from the first point to the second is from 6 meters/second to 9 meters/second, or 3 m/s. This represents the change `dv in velocity. The run is from 1.5 seconds to 3.5 seconds, or 2 seconds, and represents the change `dt in clock time. The slope, being the rise divided by the run, is 3 m/s / (2 sec) = 1.5 m/s^2. This slope represents `dv / `dt, which is the average acceleration during the time interval. You may if you wish comment on your understanding to this point.
STUDENT QUESTION
Are we going to use the terms
acceleration and average acceleration interchangeably in this course? I just
want to make sure
I understand.
INSTRUCTOR RESPONSE
Good question.
The term 'acceleration' refers to instantaneous acceleration, the acceleration at a given instant.
The term 'average acceleration' refers to the average acceleration during an interval, calculated by subtracting initial from final velocity and dividing by the change in clock time.
If acceleration is uniform, it's always the same. If acceleration is uniform, then, it is unchanging. In that case the instantaneous acceleration at any instant is equal to the average acceleration over any interval.
So when acceleration is uniform, 'acceleration' and 'average acceleration' are the same and can be used interchangeably.
Acceleration isn't always uniform, so before using the terms interchangeably you should be sure you are in a situation where acceleration is expected to be uniform.
This can be visualized in terms of graphs:
The instantaneous acceleration can be represented by the slope of the line tangent to the graph of v vs. t, at the point corresponding to the specified instant.
The average acceleration can be represented by the average slope between two points on a graph of v vs. t.
If acceleration is uniform then the slope of the v vs. t graph is constant--i.e., the v vs. t graph is a straight line, and between any two points of the straight line the slope is the same. In this case the tangent line at a point on the graph is just the straight-line graph itself.
Self-critique (if necessary):
Self-critique rating:
Question: `q009. In what sense does the slope of any graph of velocity vs. clock time represent the acceleration of the object? For example, why does a greater slope imply greater acceleration?
Your solution:
Confidence rating:
Given Solution:
Since the rise between two points on a graph of velocity vs. clock time represents the change in `dv velocity, and since the run represents the change `dt clock time, the slope represents rise / run, or change in velocity /change in clock time, or `dv / `dt. This is the definition of average acceleration.
Self-critique (if necessary):
Self-critique rating:
Question: `q010. This is the same situation as in the preceding problem: An automobile coasts down a hill with a constant slope. At first its velocity increases at a very nearly constant rate. After it attains a certain velocity, air resistance becomes significant and the rate at which velocity changes decreases, though the velocity continues to increase. Describe a graph of velocity vs. clock time for this automobile (e.g., neither increasing nor decreasing; increasing at an increasing rate, a constant rate, a decreasing rate; decreasing at an increasing, constant or decreasing rate; the description could be different for different parts of the graph).
Your solution:
Confidence rating:
Given Solution:
Your graph should have velocity as the vertical axis and clock time as the horizontal axis. The graph should be increasing since the velocity starts at zero and increases. At first the graph should be increasing at a constant rate, because the velocity is increasing at a constant rate. The graph should continue increasing by after a time it should begin increasing at a decreasing rate, since the rate at which the velocity changes begins decreasing due to air resistance. However the graph should never decrease, although as air resistance gets greater and greater the graph might come closer and closer to leveling off. Critique your solution by describing or insights you had or insights you had and by explaining how you now know how to avoid those errors.
Self-critique (if necessary):
Self-critique rating:
Question: `q011. An automobile coasts down a hill with a constant slope. At first its velocity increases at a very nearly constant rate. After it attains a certain velocity, air resistance becomes significant and the rate at which velocity changes decreases, though the velocity continues to increase. Describe a graph of acceleration vs. clock time for this automobile (e.g., neither increasing nor decreasing; increasing at an increasing rate, a constant rate, a decreasing rate; decreasing at an increasing, constant or decreasing rate; the description could be different for different parts of the graph).
Your solution:
Confidence rating:
Given Solution:
Your graph should have acceleration as the vertical axis and clock time as the horizontal axis. At first the graph should be neither increasing nor decreasing, since it first the acceleration is constant. Then after a time the graph should begin decreasing, which indicates the decreasing rate at which velocity changes as air resistance begins having an effect. An accurate description of whether the graph decreases at a constant, increasing or decreasing rate is not required at this point, because the reasoning is somewhat complex and requires knowledge you are not expected to possess at this point. However it is noted that the graph will at first decrease more and more rapidly, and then less and less rapidly as it approaches the t axis.
STUDENT QUESTION: Can you clarify some more the differences in acceleration and velocity?
INSTRUCTOR RESPONSE: ** Velocity is the rate at which position changes and the standard units are cm/sec or m/sec.
Acceleration is the rate at which velocity changes and its standard units are cm/s^2 or m/s^2.
Velocity is the slope of a position vs. clock time graph.
Acceleration is the slope of a velocity vs. clock time graph. **
STUDENT QUESTION:
In the problem it states that velocity
continues to increase even though the rate at which velocity changes decreases.
I don’t understand your the slope will decrease if this is true. I can
understand a diminish in velocity and time, but not a down turn of the slope,
which is what your solution leans to.
INSTRUCTOR RESPONSE
Your thinking is good, but you need
carefully identify what it is you're describing.
The question here concerns the acceleration vs. clock time graph, whereas most
of your comments apply to the velocity vs. clock time graph.
Under these conditions the slope of the velocity vs. clock time graph will
decrease, this will occur as long as the acceleration vs. clock time graph
decreases, regardless of whether that decrease is at a constant, an increasing
or a decreasing rate.
Self-critique (if necessary):
Self-critique rating:
If you understand the assignment and were able to solve the previously given problems from your worksheets, you should be able to complete most of the following problems quickly and easily. If you experience difficulty with some of these problems, you will be given notes and we will work to resolve difficulties.
Question: `q012. Which changes its velocity more quickly, on the average, a car which speeds up from 50 mph to 60 mph in 5 seconds, or a car which speeds up from 5 mph to 25 mph in 6 seconds?
Be sure to explain your answer.
Your solution:
Confidence rating:
Question: `q013. What do we get if we divide 40 meters by 5 meters / second?
Your answer will include a number and its units. You should explain how you got the units of your answer.
Your solution:
Confidence rating:
Question: `q014. On a graph of velocity v vs. clock time t we have the two points (5 s, 10 m/s) and (10 s, 20 m/s). What is the average slope of the graph between these points, and what does this average slope mean?
Your solution:
Confidence rating:
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.
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 divide the unit sec by the unit cm/sec, what unit do we get? Does this unit correspond to any quantity we have defined? Answer the same questions for the following calculations:
(Note: It is not uncommon that a number of students have difficulty with the algebra of units. This is largely a result of a lack of practice with fractions, which is perhaps the result of a tendency to permit overuse of calculators in secondary school.
Fortunately the fractions required to do units are not difficult. Multiplication and division of fractions is an elementary-school topic, and a little practice is all that's required. The most difficult operations with fractions are addition and subtraction of fractional quantities, and this skill is not required to do unit calculations. The simplest operations with fractions are multiplication and division, which are easily refreshed with a little practice.
Click for a synopsis of fractions as related to unit calculations.
You may also access numerous Web resources, easily accessed by searching under the headings 'Multiplication of Fractions', 'Division of Fractions', 'Powers of Fractions', etc.)
2. Give at least 3 possible units for position, and 3 possible units for clock time.
In terms of these units:
(note: it would take way too long to write out:detailed answers to all these questions in your notebook; however you can easily answer these questions in the form of sketches, diagrams and abbreviations that could then be expanded into detailed answers if you were later asked for the details)
3. If the slope of a graph is constant (i.e., the slope never changes), then what can be said about the shape of the graph?
4. What can be said about the motion of an object if the slope of its position vs. clock time graph is increasing?
What can be said about the motion of an object if the slope of its velocity vs. clock time graph is decreasing?
Questions related to text
(no questions on this assignment)
Questions related to Introductory Problem Sets
1. Give the rate-of-change definition of acceleration, and explain how this definition leads to the equivalent definition
2. Explain how to solve the relationship
for `dv.
3. Explain how to solve the relationship
for `dt.
4. The figure below is from Introductory Problem Set 2, Problem # 2.
Explain how this figure depicts the definition of aAve in terms of `dv and `dt.
Questions related to Class Notes
1. If an object accelerates uniformly from rest, moving 80 cm in 10 seconds, then what are its average velocity, change in velocity and acceleration?
2. If the velocity of an object changes at a uniform rate from 5 m/s to 13 m/s between clock times t = 7 s and t = 11 s then on this interval what is its average velocity, and what is the average rate at which its velocity changes with respect to clock time?
3. The water level in a container has values 50 cm, 30 cm, 15 cm and 7 cm at respective clock times 3 second, 7 seconds, 10 seconds and 12 seconds. What is the average velocity during each of the three corresponding intervals? What is the midpoint clock time for each interval?
Sketch a graph of average velocity vs. midpoint clock time and justify your answer to the question:
Questions/problems for University Physics Students
1. The instantaneous rate of change of A with respect to B is defined to be the limiting value, as the B interval approaches zero, of the average rate of change of A with respect to B. This definition isn't specific enough to be truly rigorous, but it will do for the moment.
What therefore would be the definition of the instantaneous rate of change of position with respect to clock time?
What would be the definition of the instantaneous rate of change of velocity with respect to clock time?
2. If the velocity of an object is given by the function v(t) = 2 t^2 - t + 3, then find the average rate of change of v with respect to t for each of the following intervals:
What do you think is the instantaneous rate of change of velocity with respect to clock time at the instant when t = 2?
3. What is the instantaneous rate of change of v with respect to t at clock time t?
