Kinematics 03 - Introduction to Equations of Motion in 1 Dimension

OPENING QUESTIONS:

1) How many sig figs are in each of the following:

A) 3.501   B) .00071   C) 000071    D) .00007100

2) Let's consider the somewhat odd case where someone measures the time for the motion of an object from 2 seconds before and 2 seconds after that object passes through a fixed point (we'll call that the origin)

Graph the following function (from t = -2 seconds to time = 2 seconds)

x = t2

Now work with your groupies to provide full sentence descriptions for the motion of that object during each principal stage of that object's motion (this may surprise you).

Now find the derivative of that object at various time periods to find the slope of that curve at those instantaneous moments.

Now interpret the slope at those moments... how does the description of your slope compare to the your earlier description of the motion of that object at those points?

Note: Sig Fig click here for descriptions and here for practice

NEWBIES: For review of motion terms and concepts, please go to Kahn Academy on motion HERE (or search that site for other help)

OBJECTIVES:  

1) I will be able to find the instantaneous velocity of an object moving in NON-UNIFORM motion after today's class.

2) I will be comfortable using the basic equations of motion after today's class.

WORDS O' TODAY:

  • DISPLACEMENT (distance & direction)
  • Distance
  • VELOCITY (speed & direction)
  • Speed (distance/time)
  • ACCELERATION (change in velocity/change in time)
  • INSTANTANEOUS VELOCITY (dx/dt)

NOTE: We will revisit the concepts of SI Units, Sig Figs and dimensional analysis over and over and over again... the sooner you get comfy with those, the better.

WORK O' THE DAY: 

Please review your conceptual and o parts of the homework and be prepared to present those to the class

I'll spin the wheel and we'll have folks put homework up on the board

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NOW FOR THE GOOD STUFF:

We are particularly interested in how an object behaves when the object is under constant acceleration in ONE DIMENSION:

Learn these, memorize 'em SOONEST. These are your new best friends:

0) vt=x

1) vf = vi +at

2) vavg = (vi + vf)/2

3*) xf = xi + vit + 1/2at2

4*) vf2 - vi2 = 2a∆x

NOTES:

  • v always denotes velocity
  • a always denotes acceleration
  • x means displacement in the x direction, we will frequently swap this with y to mean displacement in the y direction
  • the subscript i always denotes initial
  • the subscript f always denotes final
  • * these are CRITICAL

Here's an interesting question for you: Compare & Contrast the following two equations of motion:

xf = xi + vit + 1/2at2  &  yf = yi + vit + 1/2at2

 

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Answer:

Hopefully you noted that those two equations are essentially the same equation shown in two different dimensions: x being the horizontal and y being the vertical

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One of the pitfalls for first year physics students is trying to do too much work all at once. Remember, all of those equations of motion work in ONE dimension.

Obviously there are relationships between the two (and we will begin t explore those today) but just remember, don't try to do too much too soon.... work in one dimension at a time.

Let's consider basic every day motion:

1) tossing an object straight up into the air

2) dropping an object straight down and having it fall to the floor

3) an object sliding across a frictionless surface

4) tossing an object straight up in the air and then catching it on the way down, one meter above where you tossed it.

(oh and by the way, get used to looking for key words like 'frictionless', those have important meanings in physics. A frictionless surface doesn't actually exist in nature, so we make them up in order to learn a basic point-- please don't get confused with how things behave in the real world!)

First step is always to draw a sketch of the situation being discussed (do that now)

The second step is always to conceptualize, describe the motion of the object... when is it speeding up, slowing down, changing direction, at rest etc...

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Recall our two MOST IMPORTANT motion equations of motion shown here in the x direction (remember, substitute y for analyzing vertical motion)

1) xf = xi + vit + 1/2at2

2*) vf2 - vi2 = 2a∆x

Notice that equation #1 is TIME dependent. Problems that concern time tend to work well with equation 1 (but they can be a bit tricky at times)

Notice that equation 2 works very well if you are given an object's initial velocity...

Here's an interesting question, how does equation #2 become VERY interesting when dealing with our situation #1?

How does equation #1 help us with situation #4?

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Here's something CRITICAL to keep in mind:

It is very, very tempting to practice "pattern matching" to solve physics problems. In other words, a problem provides certain variables so you look for a formula that has those variables and commence to plug and chug.

In all honesty there MAY be times when you resort to that. PLEASE keep those to an absolute minimum. You'll be much, much better off if you conceptualize the problem FIRST and then go formula hunting....

(I'm just sayin')

CLASSWORK (1) : Mr W is standing in his backyard with an arrow and a 45 lb recurve bow. He lifts the bow above his head (1.98m), pulls the bow back (.75 m) and launches the arrow directly upward with an initial velocity of 17.2 m/s.

We're curious how far the arrow will fly directly upwards (ignoring air friction, wind and other such stuff... we'll keep it simple).

Oddly enough, I don't want you to solve the problem, I simply want you to set it up. So....(ok ok, you can solve it if you like)

Please setup up this problem using full wolgemuthian and our main acceleration formula. You may notice I've added some extraneous information here. Your job is to identify the useful parts and setup the problem

 

HOMEWORK :

  1. Please read sections 2.4 - 2.6.

  2. Do the worked example problem 2.6 (page 34) part A and then do part B using derivatives like we've done in class (not limits as shown in the example)

  3. Work through the quick quiz 2.6 on page 38 (that one might make your head spin a bit, that's ok, we're still learning that stuff)

  4. Do the worked example problem 2.7 (page 38) part A only

  5. Do the worked example 2.8 (page 39)

ALSO -- please take 15 - 20 minutes to ponder the following thought experiment:

A *huge* skill to learn in this class is *HOW* to learn. How to learn isn't just grinding on problem sets. How we learn is most important when you get stuck.

SPEAKING OF GETTING STUCK!!!!

Here's a problem that WILL get you stuck:

WITHOUT doing any research whatsoever (PLEASE!! This is not a chance to show off using Google, it is a REASONING exercise), analyze the following problem for no more than 20 minutes and come prepared to discuss on Monday. You are free to discuss with your classmates over a white chocolate mocha (in a mug!) but NO ONE else can assist.

Consider nuclear fission-- the process where a heavy atom (such as uranium) "splits" into smaller atoms:

1 neutron is captured by a uranium-235 atom. That uranium-235 atom contains 92 protons and 143 neutrons.

That means there are 92 protons and 144 total neutrons present after the capture of the 'invading' neutron which results in the uranium-236 atom shown in the center of that image.

That uranium-236 atom splits into two 'daughter' atoms and 3 single neutrons as shown.

Most of us believe that in this reaction, some of the mass of the atom is converted into energy using the process outlined by the famous E=mc2 equation.

However, if you look closely (if you've had chemistry you SHOULD be able to do this, if you haven't, please do take my word for it!) you'll notice there are the same number of protons and neutrons present before AND after the reaction.

In other words, none of the mass (protons and neutrons) was converted into energy. So, here's the issue. Most of us "know" that mass is converted into energy in a nuclear reaction. The graphic above clearly shows that no particle is converted into energy.... and yet the mass of a uranium atom before the reaction is greater than the mass of all the particles created by the reaction....hmmmm

Clearly our thinking needs revision (hint electrons aren't involved in any way)

So... you have the problem, you have the solution (less mass is present after the reaction) but you don't have an explanation.

Come prepared next time with a possible description of what MIGHT be going on during the fission reaction shown above.

PLEASE DO NOT CHEAT and research on the web, that totally defeats the purpose of this exercise and I'll probably know if if you do!) This is not a chemistry or physics problem per se, but (once again), an exercise in reasoning.