Practice for test

Here are some practice problems.

1.  Consider a ball dropped from rest.  It reaches a top speed of 36 m/s.

a.  From what height was it dropped?
b.  How much time did it spend in air?

2.  forthcoming.

3.  Review these ideas:

a.  SI standards - what they are, what they were, why was there a change?  You don't need to know specific numbers, but rather things like:  the meter is now based on the speed of light, though it was once based on the distance between north pole and equator.

b.  Unit conversions

c.  Odd numbers rule (Galileo)

d.  Graph and how to interpret them - think about the lab

e.  How to use all equations of motion

f.  The related demonstrations (ballistics cart, etc.)

Worth watching.

https://www.youtube.com/watch?v=Lr_bQs4oXgU


IF you have time - it is pretty long.

Class work

Folks

Sorry to be absent.

Try these problems:

How long will it take a ball released from rest to fall a distance of 25-m?  And how fast will it me traveling before it hits ground?

Why did the 2 balls (one launched and one dropped) hit simultaneously?

If a ball is launched from a height of 10-m, at a speed (horizontally) of 5 m/s, how long will it take to hot the ground?  And how far horizontally will it travel?

What is the odd numbers rule all about?

Recall the ballistics cart on the track.  Why did the balls land in the cart?

For next class, find out some biographical info about Isaac Newton.

HW

Review the equations of motion. 

Also, revisit the demonstrations you saw today:  cart that launched and dropped the ball, 2 balls projected simultaneously.  Write a brief statement or paragraph that explains what happened and why.

stuff

http://www.xamuel.com/inverse-graphing-calculator.php?phrase=LALLY

This is just a little fun -- this website will allow you to create a complicated equation based on your name.

Also - make sure you try the additional homework problems posted yesterday.

Barometer story

Angels on a Pin

A Modern Parable
by Alexander Callandra
Saturday Review, Dec 21, 1968.

Some time ago I received a call from a colleague who asked if I would be the referee on the grading of an examination question. He was about to give a student a zero for his answer to a physics question, while the student claimed he should receive a perfect score and would if the system were not set up against the student: The instructor and the student agreed to submit this to an impartial arbiter, and I was selected.

I went to my colleague's office and read the examination question: "Show how it is possible to determine the height of a tall building with the aid of a barometer."

The student had answered: "Take a barometer to the top of the building, attach a long rope to it, lower the barometer to the street and then bring it up, measuring the length of the rope. The length of the rope is the height of the building."

I pointed out that the student really had a strong case for full credit since he had answered the question completely and correctly. On the other hand, if full credit was given, it could well contribute to a high grade for the student in his physics course. A high grade is supposed to certify competence in physics, but the answer did not confirm this. I suggested that the student have another try at answering the question I was not surprised that my colleague agreed, but I was surprised that the student did.

I gave the student six minutes to answer the question with the warning that the answer should show some knowledge of physics. At the end of five minutes, he had not written anything. I asked if he wished to give up, but he said no. He had many answers to this problem; he was just thinking of the best one. I excused myself for interrupting him and asked him to please go on. In the next minute he dashed off his answer which read:

"Take the barometer to the top of the building and lean over the edge of the roof. Drop that barometer, timing its fall with a stopwatch. Then using the formula d = ½gt², calculate the height of the building.

At this point I asked my colleague if he would give up. He conceded, and I gave the student almost full credit.

In leaving my colleague's office, I recalled that the student had said he had many other answers to the problem, so I asked him what they were. "Oh yes," said the student. "There are a great many ways of getting the height of a tall building with a barometer. For example, you could take the barometer out on a sunny day and measure the height of the barometer and the length of its shadow, and the length of the shadow of the building and by the use of a simple proportion, determine the height of the building."

"Fine," I asked. "And the others?"

"Yes," said the student. "There is a very basic measurement method that you will like. In this method you take the barometer and begin to walk up the stairs. As you climb the stairs, you mark off the length of the barometer along the wall. You then count the number of marks, and this will give you the height of the building in barometer units. A very direct method."

"Of course, if you want a more sophisticated method, you can tie the barometer to the end of a string, swing it as a pendulum, and determine the value of 'g' at the street level and at the top of the building. From the difference of the two values of `g' the height of the building can be calculated."

Finally, he concluded, there are many other ways of solving the problem. "Probably the best," he said, "is to take the barometer to the basement and knock on the superintendent's door. When the superintendent answers, you speak to him as follows: "Mr. Superintendent, here I have a fine barometer. If you tell me the height of this building, I will give you this barometer."

At this point I asked the student if he really did know the conventional answer to this question. He admitted that he did, said that he was fed up with high school and college instructors trying to teach him how to think, using the "scientific method," and to explore the deep inner logic of the subject in a pedantic way, as is often done in the new mathematics, rather than teaching him the structure of the subject. With this in mind, he decided to revive scholasticism as an academic lark to challenge the Sputnik-panicked classrooms of America.

(You'll note that I tweaked the end of the story just a bit, as the last few lines don't make much sense in 2014.)

more problems

Assume no air resistance.

1.  How far can a ball fall in 5 seconds?

2.  If you want to throw a ball upwards so that it hits a window 20-m above your hand, with what speed should you throw it?  How long will it take to hit the window?  (You can solve either part first.)

3.  How does the time to fall a certain distance on the Moon compare to the time to fall the same distance on Earth?  Moon gravity is 1/6 that of Earth.

Local gravity notes

Some thoughts on the acceleration due to gravity - technically, "local gravity". It has a symbol (g), and it is approximately equal to 9.8 m/s/s, near the surface of the Earth. At higher altitudes, it becomes lower - a related phenomenon is that the air pressure becomes less (since the air molecules are less tightly constrained), and it becomes harder to breathe at higher altitudes (unless you're used to it). Also, the boiling point of water becomes lower - if you've ever read the "high altitude" directions for cooking Mac n Cheese, you might remember that you have to cook the noodles longer (since the temperature of the boiling water is lower).

On the Moon, which is a smaller body (1/4 Earth radius, 1/81 Earth mass), the acceleration at the Moon's surface is roughly 1/6 of a g (or around 1.7 m/s/s). On Jupiter, which is substantially bigger than Earth, the acceleration due to gravity is around 2.2 times that of Earth. All of these things can be calculated without ever having to visit those bodies - isn't that neat?

Consider the meaning of g = 9.8 m/s/s. After 1 second of freefall, a ball would achieve a speed of .....

9.8 m/s

After 2 seconds....

19.6 m/s

After 3 seconds....

29.4 m/s

We can calculate the speed by rearranging the acceleration equation:

vf = vi + at

In this case, vf is the speed at some time, a is 9.8 m/s/s, and t is the time in question. Note that the initial velocity is 0 m/s.  In fact, when initial velocity is 0, the expression is really simple:

vf = g t

Got it?

The distance is a bit trickier to figure. This formula is useful - it comes from combining the definitions of average speed and acceleration.

d = vi t + 0.5 at^2

Since the initial velocity is 0, this formula becomes a bit easier:

d = 0.5 at^2

Or....

d = 0.5 gt^2

Or.....

d = 4.9 t^2

(if you're near the surface of the Earth, where g = 9.8 m/s/s)

This is close enough to 10 to approximate, so:

d = 5 t^2

So, after 1 second, a freely falling body has fallen:

d = 5 m

After 2 seconds....

d = 20 m

After 3 seconds....

d = 45 m

After 4 seconds...

d = 80 m

This relationship is worth exploring. Look at the numbers for successive seconds of freefall:

0 m

5 m

20 m

45 m

80 m

125 m

180 m

If an object is accelerating down an inclined plane, the distances will follow a similar pattern - they will still be proportional to the time squared. Galileo noticed this. Being a musician, he placed bells at specific distances on an inclined plane - a ball would hit the bells. If the bells were equally spaced, he (and you) would hear successively quickly "dings" by the bells. However, if the bells were located at distances that were progressively greater (as predicted by the above equation, wherein the distance is proportional to the time squared), one would hear equally spaced 'dings."

Check this out:

Equally spaced bells:

http://www.youtube.com/watch?v=06hdPR1lfKg&feature=related

Bells spaced according to the distance formula:

http://www.youtube.com/watch?v=totpfvtbzi0

Furthermore, look at the numbers again:

0 m

5 m

20 m

45 m

80 m

125 m

180 m

Each number is divisible by 5:

0

1

4

9

16

25

36

All perfect squares, which Galileo noticed - this holds true on an inclined plane as well, and its easier to see with the naked eye (and time with a "water clock.")

Look at the differences between successive numbers:

1

3

5

7

9

All odd numbers. Neat, eh?

FYI:

http://www.mcm.edu/academic/galileo/ars/arshtml/mathofmotion1.html

gravity HW

Problems with gravity.  Assume no air resistance.

1.  An object is dropped from rest.  How far could it fall in 3 seconds?

2.  How fast would it be moving after 3 seconds?

3.  If you drop a rock from a 30-m high bridge into the water below, how long will it take to hit water?

4.  Imagine throwing a ball straight up into the air, with an initial speed of 25 m/s.  Hint:  it may be wisest to call UP positive, which makes gravity negative.

a.  How long will it take to reach apogee?
b.  How high will it rise?
c.  How long will it take for the ball to return to your hands?

Think about this

HW stuff

Remember - the lab draft is due on Wednesday.  Final lab report due Friday.

A problem to play with (based on today's new material):

Consider an object that is accelerating at 2 m/s/s*.  It starts from rest and accelerates until it reaches a speed of 24 m/s.

- how long will this take?
- how far will it go during this time?  (Think about the equation for average velocity before doing this part.)

* Note that this unit could also be:  m/s^2.

See you Wednesday!