A river 500 ft wide flows with a speed of 8 ft/s with respect to the earth. A woman swims with a speed of 4 ft/s with respect to the water.
1) If the woman heads directly across the river, how far downstream is she swept when she reaches the opposite bank?
d1 =
2) If she wants to be swept a smaller distance downstream, she heads a bit upstream. Suppose she orients her body in the water at an angle of 37° upstream (where 0° means heading straight accross, as in part (a)), how far downstream is she swept before reaching the opposite bank?
d2 =
3) For the conditions of part (b), how long does it take for her to reach the opposite bank?
t =

1)

2)  

3)

Explanation:

Given:

width of river,

speed of stream with respect to the ground,

speed of the swimmer with respect to water,

Now the resultant of the two velocities perpendicular to each other:

Now the angle of the resultant velocity form the vertical:

Now the distance swam by the swimmer in this direction be d.

so,

Now the distance swept downward:

2)

On swimming 37° upstream:

The velocity component of stream cancelled by the swimmer:

Now the net effective speed of stream sweeping the swimmer:

The  component of swimmer's velocity heading directly towards the opposite bank:

Now the angle of the resultant velocity of the swimmer from the normal to the stream:

Now let the distance swam in this direction be d'.

Now the distance swept downstream:

3)

Time taken in crossing the rive in case 1:

Time taken in crossing the rive in case 2:

The Cherokee nation was one of the "Five Civilized Tribes" in the southeast, and Andrew Jackson planned their removal along with all other tribes existing east of the Mississippi River. Chief John Ross and other leaders of the Cherokee nation wrote a letter to Congress to protest the 1835 Treaty of New Echota.

Letter from Chief John RossIt is well known that for a number of years past we have been harassed by a series of vexations, which it is deemed unnecessary to recite in detail, but the evidence of which our delegation will be prepared to furnish. With a view to bringing our troubles to a close, a delegation was appointed on the 23rd of October, 1835, by the General Council of the nation, clothed with full powers to enter into arrangements with the Government of the United States, for the final adjustment of all our existing difficulties. The delegation failing to effect an arrangement with the United States commissioner, then in the nation, proceeded, agreeably to their instructions in that case, to Washington City, for the purpose of negotiating a treaty with the authorities of the United States.
After the departure of the Delegation, a contract was made by the Rev. John F. Schermerhorn, and certain individual Cherokees, purporting to be a "treaty, concluded at New Echota, in the State of Georgia, on the 29th day of December, 1835, by General William Carroll and John F. Schermerhorn, commissioners on the part of the United States, and the chiefs, headmen, and people of the Cherokee tribes of Indians." A spurious Delegation, in violation of a special injunction of the general council of the nation, proceeded to Washington City with this pretended treaty, and by false and fraudulent representations supplanted in the favor of the Government the legal and accredited Delegation of the Cherokee people, and obtained for this instrument, after making important alterations in its provisions, the recognition of the United States Government. And now it is presented to us as a treaty, ratified by the Senate, and approved by the President [Andrew Jackson], and our acquiescence in its requirements demanded, under the sanction of the displeasure of the United States, and the threat of summary compulsion, in case of refusal. It comes to us, not through our legitimate authorities, the known and usual medium of communication between the Government of the United States and our nation, but through the agency of a complication of powers, civil and military.
By the stipulations of this instrument, we are despoiled of our private possessions, the indefeasible property of individuals. We are stripped of every attribute of freedom and eligibility for legal self-defence. Our property may be plundered before our eyes; violence may be committed on our persons; even our lives may be taken away, and there is none to regard our complaints. We are denationalized; we are disfranchised. We are deprived of membership in the human family! We have neither land nor home, nor resting place that can be called our own. And this is effected by the provisions of a compact which assumes the venerated, the sacred appellation of treaty.
We are overwhelmed! Our hearts are sickened, our utterance is paralized, when we reflect on the condition in which we are placed, by the audacious practices of unprincipled men, who have managed their stratagems with so much dexterity as to impose on the Government of the United States, in the face of our earnest, solemn, and reiterated protestations.
The instrument in question is not the act of our Nation; we are not parties to its covenants; it has not received the sanction of our people. The makers of it sustain no office nor appointment in our Nation, under the designation of Chiefs, Head men, or any other title, by which they hold, or could acquire, authority to assume the reins of Government, and to make bargain and sale of our rights, our possessions, and our common country. And we are constrained solemnly to declare, that we cannot but contemplate the enforcement of the stipulations of this instrument on us, against our consent, as an act of injustice and oppression, which, we are well persuaded, can never knowingly be countenanced by the Government and people of the United States; nor can we believe it to be the design of these honorable and highminded individuals, who stand at the head of the Govt., to bind a whole Nation, by the acts of a few unauthorized individuals. And, therefore, we, the parties to be affected by the result, appeal with confidence to the justice, the magnanimity, the compassion, of your honorable bodies, against the enforcement, on us, of the provisions of a compact, in the formation of which we have had no agency.