Rocket Science 101 -- Quiz 1

Rocket Science 101: Basic Physics of Rocketry

Rocket Science 101: Basic Math

Everything You Wanted to Know About Sound in 12 Minutes

I like it when people are smart and creative and humorous and competent and comprehensive.

Boredom in Math Class Has Some *Interesting* Results

I got bored in Diffy Q's today, so I made up a new function. It is a sort of step function, but it increases so fast that it blows your mind. I had to make up some new notation for it. Here it is:

f(x) = x!_x

What it does is it takes the number x and puts it through x number of factorials. Thus,

If x = 1, f(x) = 1! = 1
If x = 2, f(x) = 2!! = 2
If x = 3, f(x) = 3!!! = 720!
If x = 4, f(x) = 4!!!! = (6.2 x 10²³)!!

Obviously this is an impressive function. Just imagine how big a number you'd get if you have x = Googolplex^{GoogolplexGoogolplexGoogolplex}. YIPERS!!

Back to Baby Math

It's rather funny: the higher up in mathematics you go, you often revisit "duh" topics, usually to illustrate a point. Today, in my diffy q's class, copied from the blackboard verbatim, was written this:

a + x = b
-a + (a + x) = -a + b
(-a + a) + x = -a + b
0 + x = -a + b
x = b - a

ax = b
a^-1(ax) = a^-1b
(a^-1a)x = a^-1b
1x = a^-1b
x = b/a

I was chuckling inside as my professor explained about opposites and inverses and associative properties, etc.

Despite this seemingly easy revisitation to 7th grade math, this course is very difficult. Right now we're learning how to solve systems of nonhomogeneous higher order differential equations. My head is swimming with linear operators, matrix multipliers, LaPlace transforms, and superposition.

Come March 10th, I will fall into a deep sleep for 10 days.

Tidbit of the Week

Happy Pi-day!

I just figured something out. MIT is going to be posting admissions decisions at 1:59. It's pi-day. That's the pi-moment. 3/14 at 1:59. And 26 seconds.

3.141592653589793238462643383279502884197169399375105820974944592307....

The suspense is killing me!

Physics Jokes

• Artificial intelligence is no match for natural stupidity.
• Why did the chicken cross the road? Issac Newton: Chickens at rest tend to stay at rest, chickens in motion tend to cross roads. Albert Einstein: Whether the chicken crossed the road or the road crossed the chicken depends on your frame of reference.
• Q: How many theoretical physicists specializing in general relativity does it take to change a light bulb?
  
  A: Two. One to hold the bulb and one to rotate the universe.
• Does a radioactive cat have 18 half-lives?
• There has been too much action in reaction to political scandals. Please write to your congressman to repeal Newton's third law.
• Rene DesCartes walks into a bar. The bartender looks at him and says "What'll it be? Would you like a beer?" To which DesCartes replies, "I don't think--" POOF! He disappears.
• There is this farmer who is having problems with his chickens. All of the sudden, they are all getting very sick and he doesn't know what is wrong with them. After trying all conventional means, he calls a biologist, a chemist, and a physicist to see if they can figure out what is wrong. So the biologist looks at the chickens, examines them a bit, and says he has no clue what could be wrong with them. Then the chemist takes some tests and makes some measurements, but he can't come to any conclusions either. So the physicist tries. He stands there and looks at the chickens for a long time without touching them or anything. Then all of the sudden he starts scribbling away in a notebook. Finally, after several gruesome calculations, he exclaims, "I've got it! But it only works for spherical chickens in a vacuum."
  

Problems

Since school is almost all I have time for these days, I thought I'd bring it to my blog now and then.

In calculus we are doing sequences, L'Hopital's rule, and improper integrals. Pretty straight forward.

Pop quiz:
1. Is 0^0 zero or one? Or neither?
2. Limit as x approaches 1 of x^[1/(x-1)] = ?
3. The third term of an arithmetic sequence is 5 and the seventh is 19. What is the first term?
4. This is a fun problem. Brownie-points to anyone who gets it right:

A rocket, starting from rest, travels straight up with an acceleration of 58.0 m/s^2. When the rocket is at a height of 562 m, it produces sound that eventually reaches a ground-based monitoring station directly below. The sound is emitted uniformly in all directions. The monitoring station measures a sound intensity I. Later, the station measures an intensity 1/3I. Assuming that the speed of sound is 343 m/s, find the time that has elapsed between the two measurements.

Hints

In case you were curious about the previous two problems but didn't exactly know where to start, here are a few tips:

1. Start with the law of cosines. You need it for the distance between the two planes. Take the derivative of the law of cosines with respect to time. You want to know da/dt (a being the distance between the two planes).
2. First find theta one and theta two. You need to find the net force on the system, which would be the force of m₂ minus the force of m₁. You might want to draw a free body diagram for each mass, and don't forget friction. You'll need the formula for friction, and a few Newtonian mechanics equations. Good luck!

Challenging Problems

You're good if you can get these two:

1. Two planes are flying at 32,000 feet moving directly toward a point above a control tower. The first plane is 100 miles from being directly above the tower and the second is 220 miles from being directly over the tower. The angle between them is 120 degrees. If the first plane is moving at 320 mph and the second plane is moving at 400 mph, how fast is the distance between the two planes decreasing (see figure 1)?
2. A system is positioned on a ramp according to figure 2. The mass of m₁ is equal to 8.0 kg and the mass of m₂ is equal to 23.0 kg. m₁ is pulled directly upwards by a magnet with a force of 8.5 N. The coefficient of kinetic friction μ for m₁ is 0.370. μ for m₂ is 0.289. (a) Find the acceleration of the system. (b) How much time will it take for m₁ to reach the top of the ramp?

Figure 1:

(Click to enlarge)

Figure 2:

(Click to enlarge)

The latter of the two is one I made up myself. It's easy to make up physics problems... but solving them is another story.

I'll tell you right now that the first one is not solvable without knowledge of trig formulas and calculus. The second problem just requires basic trig knowledge (SOHCAHTOA), and kinematic equations.

I'll post hints sometime this week, and will hopefully have the solutions all worked out by next weekend!

Some Math Jokes

What does a college freshman who failed his first calculus test have in common with a college freshmen who got a speeding ticket going 60mph in a 30mph zone?

Neither student knew the limits.