When you observe the motion of a ball (up and down), what do you notice about the path of the ball and its speed?

When a ball moves up and down, there is a gravitational force of attraction acting downwards in the vertical direction. The frictional force of air is being ignored and the impact of the ball with the ground is considered to be a perfectly elastic collision. The force on the ball accelerates it. As the acceleration is downwards, it increases the speed of the ball when it is moving downwards and decreases the speed when it is moving upwards. The force acting on the ball is proportional to the mass of the ball and as a result the acceleration of the ball is constant.
If a ball is dropped from a height h, its initial velocity is 0. As it moves down the velocity increases. After the ball has traveled a distance d, the velocity is equal to sqrt(2*g*d) , where g is the gravitational acceleration equal to 9.8 m/s^2. The ball strikes the ground with velocity sqrt(2*g*h) ; it then starts to move up with velocity sqrt(2*g*h) . The velocity of the ball after moving up a distance d is equal to sqrt(2*g*h - 2*g*d) . At height h, the velocity of the ball is 0 and it starts to move downwards again. Under ideal conditions, the ball would continue to move up and down alternately forever.

What does Woolf speculate happened to a woman "born with a great gift" in the sixteenth century?

Woolf writes about a theoretical woman named Judith who was the talented sister of Shakespeare. She writes that "it would have been impossible, completely and entirely, for any woman to have written the plays of Shakespeare in the age of Shakespeare." While Shakespeare was educated, his talented sister would have stayed at home. While she possessed talent, it would not have been cultivated by learning. Even if she had picked up one of her brother's books, her parents would likely have told her to put it down so she could do some mending or cooking. While her parents loved her, they knew that her life would involve marrying rather than writing.
She would have been engaged to marry to a "neighboring wool-stapler" while still a teen. When she cried to her father that she did not want to marry, he might have beaten her or tried to bribe her with promises of gifts of finery. If she had escaped to London to act or write, she would have been greeted with jeers by the managers of the theaters. One actor-manager might have taken "pity" on her and impregnated her, and she would have killed herself as a result.

What is the genre of the story "After Twenty Years" by O. Henry?

O. Henry's "After Twenty Years" falls under the genre of Realism.
Set in New York City at the turn of the twentieth century, this tale of the fated reunion of two old friends after twenty years, and involves some surprising contradictions. It begins with a policeman who walks his beat by checking the doorways and the store doors. As he does so, he comes to the dark doorway of what was once a restaurant and finds a man standing there.

"It's all right, officer," he said, reassuringly. "I'm just waiting for a friend. It's an appointment made twenty years ago."

While the stranger explains, he lights a cigar. As he does so, the light reveals a scar near the man's right eyebrow and a large diamond in his necktie. These distinguishing marks identify the man as the policeman's old friend, who is now known as 'Silky Bob' and wanted by police in Chicago. When Jimmy Wells, the policeman, recognizes his old friend as a wanted man, he does not have the heart to arrest the ill-fated man. After ascertaining that Bob will allow his friend thirty minutes before he leaves the doorway, Wells returns to the police station and asks a plainclothes policeman if he will make the arrest for him as he does not have the heart to do so.
In the end, while there is the ironic and fateful meeting of policeman Jimmy Wells and the wanted man 'Silky Bob,' his old friend, such a happenstance is not completely unrealistic.

f(x) = 1 / (x^2+1) , (1, 1/2) Set up and evaluate the definite integral that gives the area of the region bounded by the graph of the function and the tangent line to the graph at the given point.

We first need to find the equation of the tangent line. We will do that using the following formula:
y=y_1+f'(x_1)(x-x_1)
where (x_1,y_1) is the point where the tangent touches the graph.
First we calculate the derivative.
f'(x)=(0cdot(x^2+1)-1cdot2x)/(x^2+1)^2=-(2x)/(x^2+1)^2
f(1)=-2/2^2=-1/2
Now we can calculate the equation of the tangent line.
y=1/2-1/2(x-1)
y=1/2-1/2x+1/2
y=1-1/2x
We have to determine the bounds of integration. To do that, we need to know the where the tangent line intersects the graph of the function (point A in the image below). Hence, we need to solve the following system of equations.
y=1/(x^2+1)
y=1-1/2x
1-1/2x=1/(x^2+1)
Multiply by 2(x^2+1). 
2x^2+2-x^3-x=2
-x^3+2x^2-x=0
Factoring yields.
-x(x^2-2x+1)=0
From this we see that the solutions are x_1=0, x_(2,3)=1.
Therefore, the bounds of integration will be 0 an 1. To set up the integral we only need to look at the image below and see that the tangent line lies below the graph of the function which means we need to subtract it from the function. Alternatively, we could switch the bounds of integration or simply take absolute value from the whole integral.
A=int_0^1 (1/(x^2+1)-1+1/2x)dx=
(arctan x-x+x^2/4)|_0^1=
pi/4-1+1/4-0+0-0=1/4(pi-3)
The are of the region bounded by the graph of the given function and tangent to that function at (1,1/2) is equal to 1/4(pi-3).                                                                                        

What is the insight of the story of Oedipus?

The Greeks often dealt with the quality of hubris, especially in their tragedies. For them, it meant excessive, even overweening, pride, which more than likely resulted in defiance of the gods; in ancient Greek, hubris led to nemesis, and the root of nemesis means "given what is due." In other words, the quality of hubris leads to one's downfall and utter ruin—what other punishment could there be for a characteristic that runs counter to, even leading to a disrespect of, the gods' will?
When Oedipus visited the Delphic oracle and heard the prophecy that he would kill his father and marry his mother, he heard directly from Apollo's mouthpiece. In attempting to outwit the oracle (and his fate), Oedipus decides not to return to his home in Corinth but, rather, to head toward Thebes; this way, he could not possibly fulfill the prophecy, he thinks, because his parents live in Corinth.
He does not know that he was adopted by them, having been given up for dead by his mother, the queen of Thebes, upon his birth. He has such hubris that he thinks he can simply avoid his fate—challenging the oracle and the god, Apollo—despite the fact that he is a mere mortal. If it isn't excessive pride to believe that, with one move, Oedipus can avoid his fate and reject Apollo's prophecy, then I don't know what is. The play warns its audience to avoid the quality of hubris and to respect the gods' will.

The Greek playwrights of the classical period intended to teach their audiences with their staged dramas. Sophocles is a part of this tradition and imparts an important universal message through Oedipus Rex. The insight he offers is that unchecked pride can lead one to catastrophic results, especially if it leads one to challenge the power of the gods or of fate.
Oedipus learns at the start of the play that he can cure the plague that is afflicting Thebes by solving the murder of Laius. Because Oedipus has saved the people of Thebes once before by solving the riddle of the Sphinx, he is sure he can solve this problem as well, and he calls for the blind prophet Tiresias to question him on the matter. Tiresias hesitates to inform Oedipus of what he knows, and this entire opening sequence reflects Oedipus’s pride. He believes he is more than able to save his people, and because he is so sure he misses the signs that something is amiss. He even accuses his brother-in-law and Tiresias of conspiring against him as a result of his pride.
Later we learn the truth: Oedipus himself had killed his father, Laius, in an attempt to avoid a prophecy that he would murder his father and marry his mother. Years before, he had left his parents (who were, unknown to him, adoptive parents) and moved to Thebes. By killing Laius, and marrying Jocasta, he fulfills the prophecy he tried to avoid. Thus, the second part of the universal message becomes clear: man should not attempt to challenge fate or the power of the gods. His pride led him to cross this line, and it led to the catastrophic ending.

How does the other Wes Moore tell his family he earned enough money for the sneakers?

Wes, the other Wes Moore of the title, has grown up in a broken home. He doesn't meet his father until he's eight years old and his older brother Tony's involved in selling drugs. There's a terrible sense of inevitably when Wes himself starts getting mixed up in the drug business at the age of twelve.
Growing up in the projects, Wes is acutely aware that the only people who seem to have anything are those who commit crime for a living. He envies Tony's fancy wardrobe even though he's just wound up in hospital after being shot during a drug deal gone wrong. Despite his brother's injuries, Wes makes the fateful decision to follow in his footsteps and start dealing drugs.
One day, Tony notices that Wes has a large collection of expensive sneakers. He asks him how he was able to afford them. Wes responds by saying that he got the money to buy the sneakers from working as a DJ; this is the same lie he told his mother. But Tony's not stupid; he knows where his kid brother really got the money from. Tony may be involved in dealing drugs himself, but it's clear that he wants Wes to have a better life, and not to follow the same dangerous path in life that he chose for himself.

How would you compare and contrast the two poems "America" by Claude McKay and "The New Colossus" by Emma Lazarus? What are these poems’ views of the American experience? What similarities exist between the poems? How do the narrators’ perspectives differ? What literary elements do the authors use to assist in conveying their particular perspective?

Emma Lazarus's "The New Colossus" and Claude McKay's "America" both speak about the promise of America, but McKay has a cynical view of what America can offer him, while Lazarus is idealistic about America's promise. 
Lazarus's poem, which she wrote in 1883 to raise funds to construct the Statue of Liberty, uses the symbol of Lady Liberty to stand for American values of freedom. She contrasts the Statue of Liberty with the Colossus at Rhodes, an ancient statue that stood for conquest. Unlike the Colossus, the Statue of Liberty stands at our "sunset gates," a reference to the Western world, as the "Mother of Exiles." Lazarus also personifies Liberty, stating,

From her beacon-hand
Glows world-wide welcome; her mild eyes command
The air-bridged harbor that twin cities frame.

Liberty is not merely a statue but a powerful force who sends out messages of welcome across the world and commands the harbor between Brooklyn and Manhattan (which were then "twin cities"). In the second stanza, Lady Liberty calls for the world to send her "Your huddled masses yearning to breathe free." Liberty, a symbol of American freedom, in contrast to the Old World (what Lazarus refers to as "ancient lands"), welcomes the poor and oppressed of the world with open arms.
McKay's poem, like Lazarus's poem, personifies the United States as a woman; however, his America, unlike Lazarus's, is the type of cruel person who "feeds me bread of bitterness." She, unlike the fair and welcoming Lady Liberty of Lazarus's poem, has a "tiger's tooth" and takes away "my breath of life." Although he describes the United States as "hell," McKay says he loves the United States, just like Lazarus. McKay praises the country's energy in a series of similes, such as "her vigor flows like tides into my blood" and "her bigness sweeps my being like a flood." In these similes, he compares American energy to tides and her size to a flood. He then compares himself, in a simile, to a rebel who stands before a king without any trace of ill will. Then, in an extended metaphor (and a simile in the last line), McKay compares the United States's greatness to marble statues who sink into the sands of time. While he acknowledges the United States's greatness, McKay ends his poem on a troubling note, as he questions what America's future will hold. Lazarus, on the other hand, presents a view that is hopeful about the future of the United States.