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course PHY 201
005. Uniformly Accelerated MotionPreliminary 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.
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Question: `q001. Note that there are 13 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)
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Your solution:
The velocity changes by 20 m/s. The average acceleration is 5 m/s^2. The average velocity is 15 m/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).
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Self-critique (if necessary): ok
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Self-critique rating: ok
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Question: `q002. How far does the object of the preceding problem travel in the 4 seconds?
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Your solution:
The average velocity is 15 m/s for four seconds, so the object traveled 60 m.
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.
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Self-critique (if necessary): ok
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Self-critique rating: ok
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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.
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Your solution:
The acceleration is the change in velocity divided by the change in time. If we average the velocities and multiply that average by the time it took, it give us the distance.
confidence rating #$&*:
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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.
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Self-critique (if necessary): ok
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Self-critique rating: ok
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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.
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Your solution:
a = (vf - v0) / `dt
(v0 + vf) / 2 = vAve
((v0 + vf) / 2) * `dt = `ds
Confidence rating: 2
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.
STUDENT SOLUTION (mostly but not completely correct)
vAve = (vf + v0) / 2
aAve = (vf-v0) / dt
displacement = (vf + v0)/dt
INSTRUCTOR RESPONSE
Displacement = (vf + v0)/dt is clearly not correct, since greater `dt implies greater displacement. Dividing by `dt would give you a smaller result for larger `dt.
From the definition vAve = `ds / `dt, so the displacement must be `ds = vAve * `dt. Using your correct expression for vAve you get the correct expression for `ds.
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Self-critique (if necessary): ok
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Self-critique rating: ok
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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?
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Your solution:
The initial velocity of 5 m/s occurs at t = 0s. This makes the point on the graph ( 0 s, 5 m/s). The final velocity of 25 meters/second happens after a time interval of `dt = 4 seconds. This makes the point on the graph ( 4 s, 25 m/s).
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).
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Self-critique (if necessary): ok
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Self-critique rating:ok
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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?
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Your solution:
The graph is a straight line. It increases at a constant rate from the point (0, 5 m/s) to the point (4 s, 25 m/s).
confidence rating #$&*:
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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).
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Self-critique (if necessary): ok
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Self-critique rating: ok
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Question: `q008. What is the slope of the graph between the two given points, and what is the meaning of this slope?
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Your solution:
(25 - 5) / (4 - 0) = 5 m/s^2
confidence rating #$&*:
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Given Solution:
The rise of the graph is from 5 m/s to 25 m/s and is therefore 20 meters/second, which represents the change in the velocity of the object. The run of the graph is from 0 seconds to 4 seconds, and is therefore 4 seconds, which represents the time interval during which the velocity changes. The slope of the graph is rise / run = ( 20 m/s ) / (4 s) = 5 m/s^2, which represents the change `dv in the velocity divided by the change `dt in the clock time and therefore represents the acceleration of the object.
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Self-critique (if necessary): ok
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Self-critique rating: ok
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Question: `q009. The graph forms a trapezoid, starting from the point (0,0), rising to the point (0,5 m/s), then sloping upward to (4 s, 25 m/s), then descending to the point (4 s, 0) and returning to the origin (0,0). This trapezoid has two altitudes, 5 m/s on the left and 25 m/s on the right, and a base which represents a width of 4 seconds. What is the average altitude of the trapezoid and what does it represent, and what is the area of the trapezoid and what does it represent?
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Your solution:
The two altitudes are 5 meters/second and 25 meters/second, and their average is 15 meters/second.
The area of the trapezoid is 15 m/s * 4 s = 60 meters
confidence rating #$&*:
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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.
STUDENT COMMENT
I understand how to find the average altitude and multiply it by the amount of seconds. I also understand how to find the area of the trapezoid. But, again I don’t understand what it repreents, which is the product of the average velocity and the time interval, or the displacement.
INSTRUCTOR RESPONSE
If you multiply the average velocity on a time interval by the duration of the interval, you get the displacement.
Since the average altitude represents the average velocity and the width represents the duration of the time interval, the product therefore represents the displacement.
Since the product of average altitude and width is area, it follows that this product represents the displacement.
Self-critique (if necessary): ok
Self-critique rating: ok
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Question: `q010. Students at this point often need more practice identifying which of the quantities v0, vf, vAve, `dv, a, `ds and `dt are known in a situation or problem. You should consider running through the optional supplemental exercise ph1_qa_identifying_quantities.htm . The detailed URL is http://vhcc2.vhcc.edu/dsmith/genInfo/qa_query_etc/ph1/ph1_qa_identifying_quantities.htm If you are able to quickly identify all the quantities correctly 'in your head', the exercise won't take long and it won't be necessary to type in any responses or submit anything. If you aren't sure of some of the answers, you can submit the document, answer and/or asking questions on only the problems of which you are unsure.
You should take a quick look at this document. Answer below by describing what you see and indicating whether or not you think you already understand how to identify the quantities. If you are not very sure you are able to do this reliably, indicate how you have noted this link for future reference. If you intend to submit all or part of the document, indicate this as well.
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Your solution:
I think I am able to do this. I have bookmarked the link.
Confidence rating: 3
Given Solution:
You should have responded in such a way that the instructor understands that you are aware of this document, have taken appropriate steps to note its potential usefulness, and know where to find it if you need it.
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Self-critique (if necessary): ok
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Self-critique rating: ok
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.
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Question: `q011. The velocity of a car changes uniformly from 5 m/s to 25 m/s during an interval that lasts 6 seconds. Show in detail how to reason out how far it travels.
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Your solution:
vAve=`ds/`dt
vAve=(5 m/s+25 m/s)/2=15 m/s
15 m/s*6 s=90 m
confidence rating #$&*:
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Question: `q012. The points (5 s, 10 m/s) and (10 s, 20 m/s) define a 'graph trapezoid' on a graph of velocity vs. clock time.
What is the average 'graph altitude' for this trapezoid?
Explain what the average 'graph altitude' means and why it has this meaning.
What is the area of this trapezoid? Explain thoroughly how you reason out this result, and be sure to include and explain your units.
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Your solution:
average `graph altitude`=(10 m/s+20 m/s)/2=15 m/s
The average graph altitude is calculated by taking to two velocity points of 10 m/s and 20 m/s and dividing it by 2 to find the average. The average graph altitude represents the average velocity during the time interval.
area of trapezoid=average altitude*base(time interval)
area of trapezoid=(15 m/s)*5 s=75 m
confidence rating #$&*:
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Question: `q013. On a certain interval of duration `dt an object has initial velocity v_0 and final velocity v_f. In terms of the symbols v_0, v_f and `dt, what are the values of the following?
•vAve
•`dv
•`ds
•aAve
Be sure to explain your reasoning.
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Your solution:
vAve=(v_0+v_f)/2
`dv=v_f-v_0
`ds=((v_0+v_f)/2)*`dt
aAve=`dv/respect to clock time
confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Self-critique (if necessary): ok
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Self-critique rating: ok
@&
`dv isn't one of the given quantities, though of course you defined it a couple of lines above.
'respect to time' is not a given quantity either, though its use here is otherwise appropriate and insightful.
aAve = (vf - v0) / `dt.
*@
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Self-critique (if necessary):
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Self-critique rating:
&#Your work looks good. See my notes. Let me know if you have any questions. &#