Worth watching

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


IF you have time - it is pretty long.

A block HW

In addition to working on the test prep stuff below, look into these questions:

- What is Galileo's "Siderius Nuncius" and what does it contain?  Hint:  look for images.

- What is so controversial about Galileo's "Dialogue"?

test practice

Test is next week.  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?
c.  Draw 3 related graphs:  d vs. t, v vs. t, a vs. t.

2.  Consider a ping pong ball gun that shoots a ping pong ball horizontally at 10 m/s.  It is mounted on a desktop, 1.2-m above the floor.  Where (horizontally) should you place a cup to catch the ball?

3.  A soccer ball is kicked at a 20-degree angle, with an initial speed of 15 m/s.

a.  How far will it travel (horizontally)?
b.  How long will it be in air?
c.  How high (max) will it rise above the ground?
d.  What is the other angle that would yield the same range?

4.  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.  The weirder problems (drowsy cat, problems with quadratics)

e.  Difference between distance and displacement, speed and velocity

f.  How to use all equations of motion

g.  How and when to use horizontal and vertical components of motion

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

HW

E and A - try this problem.  Consider a ball kicked at a 40-degree angle, with an initial speed of 38 m/s.  Find:

- initial components of velocity

- time in air

- max horizontal displacement

- max vertical displacement

- the other angle that would give the same horizontal range

HW and upcoming test

A -- practice the problems you've already seen.  Have a look at those we've already done, and review your notes.

E -- two things:

1.  Review problem.  A ball is released horizontally with an initial velocity of 15 m/s.  If it lands 22-m m (horizontally) from the launch point, from what vertical height was it launched?

2.  Think about how you would incorporate angles into a projectile problem?  In other words, if a ball was kicked with a speed of 25 m/s, but with an angle of 30-degrees (with respect to the ground), how would you deal with this?  How would you find time in air, horizontal displacement, max vertical displacement?  You probably won't be able to fully solve this yet, but that's ok -- think about the things you'd need to know, or ways to approach the problem.

BOTH CLASSES:

Expect a test the week after next.

Fun.

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

Local gravity, in gory detail.

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