ph1 asst 2 supplementary questions

These questions have two purposes:

• The first purpose is to help clarify how to identify positions, velocities, clock times, accelerations, and changes in these quantities for an object's motion on an interval.
• The second purpose is to understand and apply the definitions of average velocity and average acceleration.

This exercise is optional, but is strongly recommended if at this point you find you are having trouble with either or both of these tasks.

Students at this point in the course frequently confuse positions with velocities or vice versa, velocities with accelerations or vice versa, average velocity with change in velocity, average rate of change of velocity with average velocity, etc..

You have four tools to help you distinguish them from one another.  The tools are experience, definitions, descriptive phrases and units.

To think about motion on an interval, the quantities you need to identify and distinguish are

• change in position `ds
• change in clock time `dt
• initial velocity v0
• final velocity vf
• average velocity vAve
• change in velocity `dv
• average rate of change of position with respect to clock time
• average rate of change of velocity with respect to clock time
• average acceleration aAve

Experience

You have lots of experience with motion, and you can use your experience to help you sort out these quantities.  All you need to do is think about something that's moving, then focus your attention on an interval between two events in it motion.  If you can identify and hopefully estimate the quantities

Definitions

The definitions that connect the various quantities are:

Average rate of change of A with respect to B = (change in A) / (change in B).

Average velocity is average rate of change of position with respect to clock time.

Average acceleration is rate of change of velocity with respect to clock time.

Descriptive phrases

Descriptive phrases are typically not precise enough to qualify as definitions.  However they often describe the main idea and can be helpful in first understanding a concept.  Often a definition consists of a clarification of the idea conveyed by a descriptive phrase.

Here are four descriptive phrases closely related to the main definitions:

Rate of change of one quantity with respect to another is how much the first quantity changes, per unit of change in the second.

Speed is how fast something is moving.

Velocity is how fast it's moving and in what direction.

Acceleration is sort of like how quickly the speed is changing, but the better definition is how quickly the velocity is changing and in what direction it's changing.  To master this one you've got to do some real thinking and solve a variety of problems.

Units

If I asked you how fast something is moving, which could possibly be answers to the question:  50 dollars, 5 seconds, 90 pounds, 30 miles/hour, 12 meters, 20 000 barrels, 400 centimeters / second, 10 meters / second^2, 9 kilometers?

It should be clear that some of these quantities are completely irrelevant to the question.  We can eliminate 50 dollars, 20 000 barrels and 90 pounds right away.  These quantities have nothing to do with motion.

30 miles/hour might be the most obvious quantity that represents 'how fast'.  We're used to thinking in terms of speed in miles / hour.  More generally 'how fast' is answered by a quantity that tells us how far and how long it takes to go that far.  Another quantity that could answer 'how fast' is 400 centimeters / second.

5 seconds could answer the question 'how long does it take', but it doesn't tell us how fast.  12 meters, or 9 kilometers, might tell us how far, but it doesn't tell us how long it takes to go that far. 

10 meters / second^2 can be a little confusing, because it does mention meters (how far) and seconds (how long does it take), but a velocity doesn't involved a squared second.  The unit 10 meters / second^2 is what we would get if we divided the velocity unit meters / second by the time unit seconds.  We will deal with that soon enough.  For right now understand that a quantity with second^2 in the denominator is not a velocity or a speed.

Note on metric vs. nonmetric units:

• Metric units are better defined and since they are based on powers of 10, they are easier to convert.  Arithmetic is also simpler with metric units.  For most purposes there's nothing inherently wrong with non-metric units, which can easily be converted to metric units, and can in some cases be very useful.

A summary of the units used for time, position, velocity and acceleration:

• Clock times or changes in clock time (i.e., time intervals) are expressed in units of time.  The standard unit of time is the second.  For different applications, it's more convenient to express clock times and time intervals in other units such as nanoseconds, microseconds, milliseconds, minutes, hours, years, centuries, millenia, etc..
• Positions or change in positions (i.e., displacements) are expressed in units of length or distance.  Typical metric units for position are meters, centimeters, millimeters and kilometers.  Other commonly used units are feet, inches, yards and miles. 
• Velocities and changes in velocity are expressed in units of length or distance, divided by units of time.  Typical metric units for velocity are meters / second, centimeters / second, kilometers / second.  Other commonly used units are feet / second and miles / hour.
• Accelerations are expressed in units of velocity, divided by units of time.  Typical metric units for velocity are (meters / second) / second and (centimeters / second) / second.  Other units might include (miles / hour) / second, (kilometers / hour) / second, (feet / second) / second, (kilometers / hour) / year. 

To avoid mistaking quantities, you should always take a little time to verify unit.  Double-check to be sure that a quantity you have identified as, say a position, actually has the appropriate units for a position. 

• If a quantity does not have units of time, then it isn't a clock time, it isn't a time interval, it isn't a change in clock time.
• If a quantity does not have units of position, then it isn't a position or a change in position.
• If a quantity does not have units of velocity, then it isn't a velocity or a change in velocity.
• If a quantity does not have units of acceleration, then it isn't an acceleration.

Example:  Ball tossed upward

Imagine that I throw a ball straight up into the air.  We can choose to focus our attention on any interval of the ball's motion:

• We can choose to think about the interval between its release from my hand and the instant it first contacts the ground. 
• Or we can choose to think about the interval between its release from my hand and the instant it reaches its highest point. 
• Or we can think about the interval corresponding to the first two seconds of its motion, starting the interval at the instant of release. 
• Or we can think about the interval between its release from my hand and its first contact with the ceiling, provided I have the bad judgement to do this indoors.

Each of these intervals is characterized in terms of various positions, velocities and accelerations.

`q001.  You are asked in this question to make some estimates.  Don't spend a lot of time trying to make very accurate estimates.  Any reasonable estimate of any of these quantities will be fine.

Consider the example of the ball on the interval between release and first contact with ceiling.  We will agree to measure position in the vertical direction, from the floor.  Imagine yourself in a room of your choosing, tossing the ball upwards.  Assuming you are 1.5 meters tall, the position of your feet would be on the floor, at position 0 meters (remember position is measured from the floor).  The top of your head would be at position 1.5 meters.  Your knee would probably be at position .5 meters.   A typical ceiling in a residence is about 2.5 meters above the floor.

• If you were tossing the ball into the air, at what position relative to the floor do you think the ball would be released, and what would be its position at first contact with the ceiling?  It really doesn't matter what position you choose, but it would probably be somewhere between your knees and the top of your head.

Your answer:

 

Your confidence rating:

• If you can throw a ball 15 feet high (the average person can easily so so), you can throw it upward at 10 meters / second.  If you can throw a ball 60 feet high (the average person can't) they you can throw it upward at 20 meters / second.  It just barely get it to the ceiling you would need to throw it at about 5 meters / second.  So pick a speed between 10 and 20 meters / second.  At what speed do you estimate you might toss the ball upward?

Your answer:

 

Your confidence rating:

• How long do you think it would take the ball to reach the ceiling?  For comparison, a ball would require about 0.6 seconds to fall from a typical ceiling to the floor.

Your answer:

 

Your confidence rating:

• No matter how fast you throw the ball upward, once it leaves your hand it will begin slowing immediately, due to the influence of gravity.  For every tenth of a second between release and its first contact with the ceiling, it will slow by about one meter / second.  By how much do you think the ball in your example would slow before reaching the ceiling?  Don't do any detailed calculations.  Just make an estimate you think is reasonable.  Of course if you are so inclined, you can do a simple mental calculation but don't spend over a minute or two trying to get an accurate answer.

Your answer:

 

Your confidence rating:

• How fast do you think the ball would be traveling when it first contacts the ceiling (the answer isn't 0.  Typically the ball will quickly be stopped by the ceiling, but that happens during a short time interval after it first contacts the ceiling.  Up to the instant of first contact, the only thing that has slowed the ball is gravity.

Your answer:

 

Your confidence rating:

• Now, identify each of the following quantities for the specified interval, as you have estimated them.  There will very likely be some inconsistencies among your quantities, but don't worry about that.   Explain how you have identified each quantity , and give its numerical value and its units:  (symbols)

The ball's initial and final positions.

Your answer:

 

Your confidence rating:

The ball's initial and final velocity:

Your answer:

 

Your confidence rating:

The change in the ball's position:

Your answer:

 

Your confidence rating:

The change in the ball's velocity:

Your answer:

 

Your confidence rating:

The change in clock time.

Your answer:

 

Your confidence rating:

• Based on your answers to the preceding, find the values of the following:

The ball's average velocity.

Your answer:

 

Your confidence rating:

The change in the ball's velocity.

Your answer:

 

Your confidence rating:

The average rate of change of the ball's position with respect to clock time, based on the definition of average rate of change of A with respect to B.  Be sure to explain how you have applied the definition of average rate, clearly identifying your A and B quantities and applying the definition step by step.

Your answer:

 

Your confidence rating:

The average rate of change of the ball's velocity with respect to clock time, based on the definition of average rate of change of A with respect to B.  Be sure to explain how you have applied the definition of average rate, clearly identifying your A and B quantities and applying the definition step by step.

Your answer:

 

Your confidence rating:

The following quantity isn't used much in analyzing motion in a first-semester physics course, though it does become important in more advanced courses.  However it can easily be reasoned out from the quantities you have determined to this point, and it provides another exercise in applying the average rate:

The average rate of change of the ball's velocity with respect to its position, based on the definition of average rate of change of A with respect to B.  Be sure to explain how you have applied the definition of average rate, clearly identifying your A and B quantities and applying the definition step by step.

Your answer:

 

Your confidence rating:

• List the value of following quantities, as you have estimated them in your previous responses:  v0, vf, `dt, a_Ave, v_Ave, `ds. `dv.  Note that a_Ave is the average acceleration, which is defined as the average rate of change of velocity with respect to clock time.

Your answer:

 

Your confidence rating:

`q002.   Make up a situation of your own.  Decide what is moving, and over what interval you wish to identify your quantities.  Then list the following, including the units of each:

The object that is moving.

Your answer:

 

Your confidence rating:

The event that begins the interval:

Your answer:

 

Your confidence rating:

The change in the object's position during the interval:

Your answer:

 

Your confidence rating:

The object's velocity at the beginning of the interval:

Your answer:

 

Your confidence rating:

The change in clock time from the beginning of the interval to the end:

Your answer:

 

Your confidence rating:

Based on your estimates, use the definitions of average velocity and average acceleration to determine each of the following.  For each question, state the definition, identify the quantities required to apply the definition, then find the quantity:

The object's average velocity on your chosen interval.

Your answer:

 

Your confidence rating:

The object's average acceleration on your chosen interval.

Your answer:

 

Your confidence rating:

Also answer the following

Is the object's average velocity equal to the average of its initial and final velocities (if you used independent estimates as instructed, it is possible, but unlikely, that the two will be the same; either way it's OK, but do the calculations to verify whether the two are or are not the same)?

Your answer:

 

Your confidence rating:

Question #2 may be repeated as many times as you wish, using different examples each time.