If the universe is infinite, does it have a center? Discuss.

Show that the power produced by a wind turbine is proportional to the cube of the wind velocity and to the square of the blade span diameter.

The Stefan-Boltzmann equation describes how much energy every object in the universe radiates away. But in the universe, some objects are warming up, while others are cooling down. How can this be?

Find a basis and calculate the dimension of the following subspaces of

$$\mathbb { R } ^ { 4 }$$

. (a) span

$$\{ ( 1 , - 1,2,0 ) , ( 2,3,0,3 ) , ( 1,9 , - 6,6 ) \}$$

. (b) span

$$\{ ( 2,1,0 , - 1 ) , ( - 1,1,1,1 ) , ( 2,7,4,1 ) \}$$

. (c) span

$$\{ ( - 1,2,1,0 ) , ( 2,0,3 , - 1 ) , ( 4,4,11 , - 3 ), ( 3 , - 2,2 , - 1 ) \}$$

, (d) span

$$\{ ( - 2,0,3,1 ) , ( 1,2 , - 1,0 ) , ( - 2,8,5,3 ) ( - 1,2,2,1 ) \}$$

.

Simplify the complex fraction.

$$\frac { \frac { y } { x } - \frac { x } { y } } { \frac { 1 } { x } + \frac { 1 } { y } }$$

Decide whether the statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain clearly.
The center of the universe is more crowded with galaxies than any other place in the universe.

how much energy is this in calories?

Explain how electromagnetic radiation helps scientists study the universe.

Give an explicit description of the span of S. $S=\left\{x,(x+1)^{2}, x^{2}+3 x+1\right\}$

Locate the ventricles in the brain.

What is the age of the universe expressed in units of the Planck time?

John drove to a distant city in 5 hours. When he returned, there was less traffic, and the trip took only 3 hours. If he drove 26 mph faster on the return trip, how fast did he drive each way?

The early Universe was dense with gamma-ray photons of energy $\sim k_{\mathrm{B}} T$ and at such a high temperature that protons and antiprotons were created by the process $\gamma \rightarrow \mathrm{p}+\overline{\mathrm{p}}$ as rapidly as they annihilated each other. As the Universe cooled in adiabatic expansion, its temperature fell below a certain value, and proton pair production became rare. At that time slightly more protons than antiprotons existed, and essentially all of the protons in the Universe today date from that time. (a) Estimate the order of magnitude of the temperature of the Universe when protons condensed out. (b) Estimate the order of magnitude of the temperature of the Universe when electrons condensed out.

Suppose that, instead of two types of electrically charged particles in the universe, there was only one type. What would you expect to happen to the structure of the universe if the particles produced (a) an attractive interaction and (b) a repulsive interaction?