2) Consider the Lorentz transformation that maps (x,t) into (x',t'), where t and x are both
in time units (so x is really x/c) and all speeds are in c units also.
a) Show that the inverse of the Lorentz transformation at v is the same as the Lorentz
transformation at -v.  Why is that required?
b) Show that x^2 - t^2 = x'^2 - t'^2, i.e., that x^2 - t^2 is invariant under Lorentz transformation

If the function u (x, y) is a harmonic conjugate of v (x, y) prove that the curves u (x, y) = st. and v (x, y) = stations. are orthogonal to each other. These curves are called level curves. Now consider the function f (z) = 1 / z
defined throughout the complex plane except the beginning of the axes. Draw them
level curves for the real and imaginary part of this function
and notice that they are two families of curves perpendicular to each other.

Consider a random variable Q with a uniform probability density function given as UQ(-1,1) and X=2Q^2+2 a) Find and plot the probability density function (pdf) of X b) Find and plot the cumulative distribution function (cdf) of X c) Find expected value and variance of X (E[X] and V[X]) d) Find expected value and variance of Q (E[Q] and V[Q]) e) Find the correlation of Q and X

.

A small bullet of mass m=0.24kg and speed v=17.8m/s embeds in a block of mass M=0.75kg suspended by a massless string of length L, after a collision as shown in the figure. If the bullet will appear on the other side of the block M with a speed v'=5m/s, instead of being embedded in it, find the maximum height the block M can reach. (Take g=9.81 m/s²). Express your answer using two decimal places.

Read the case of Youngstown Sheet and Tube Company v. Sawyer and answer the following questions. Be precise and articulate. Use proper grammar and complete sentences. There is no word count for this part but BE THOROUGH IN YOUR ANSWERS. CHECK TECHNICAL WRITING ELEMENTS, too. USE COMPLETE SENTENCES.

• What was the constitutional question in Youngstown Sheet and Tube Company v. Sawyer (1952)?
• How did the Court rule?
• How does this case illustrate the constitutional principles of separation of powers?

According to a report from Microsoft, 24% of PCs worldwide are not adequately protected by antivirus software.19 Suppose 200 PCs from around the world are selected at random.
a. Find the distribution of the sample proportion of PCs that
are not adequately protected.
b. Find the probability that the sample proportion is less
than 0.20.
c. Find the probability that the sample proportion is more
than 0.29.
d. Find a value v such that the probability the sample
proportion is less than v is 0.01.

You have a 40- lightbulb and a 100-W bulb.

1. Find the resistances of the 40-W bulb. Take that ΔV = 120 V.

2. Find the resistances of the 100-W bulb. Take that ΔV = 120 V.

3. If the two bulbs are connected separately, which of them will be brighter?

4. If the two bulbs are connected in series to a battery source, which of them will be brighter?

5. If the two bulbs are connected in parallel to a battery source, which of them will be brighter?

1. (a) What drives current? (b) What impedes current? (c) How does Ohm’s Law relate these quantities?
2. Why isn’t Ohm’s Law universally valid?
3. What is the resistance of an automobile headlight through which 3.50 A flows when 12.0 V is applied to it?
4. If a battery supplies energy to a circuit, what does the resistor do with that energy?
5. Calculate the effective resistance of a pocket calculator that has a 6.2 V battery and through which 0.20 mA flows.

IV) Many enzymes obey simple Michaelis-Menten kinetics, which are summarized
by the equation:
rate = (V max [S])/([S]+K m ); where V max = maximum velocity, [S] = concentration
of substrate; and K m = the Michaelis constant.
It is instructive to plug in a few values of [S] into the equation to see how rate is
affected. What are the rates for [S] equal to zero, equal to K m , and equal to infinite
concentration? Show all work.

using the Oyez audio file and transcript (Links to an external site.)Links to an external site. (go to "Opinion Announcement," which is a seven minute audio clip on the court's decision on the Snyder v. Phelps case), briefly summarize the facts of the Snyder v. Phelps (2011) case. By using facts presented, clearly explain why the Supreme Court ruled in favor of the Westboro Baptist Church. Then, tell us why you agree/disagree with the Court’s decision.