Question

Background:

Jerome and Paul are competitive brothers. They live on a small farm on the northern bank of a river that runs purely east and west and that flows to the east at a rate of 1.400  m/s .

The brothers have run some time trials on the farm pond, and they know that, in still water, Jerome can paddle the family canoe at a steady rate of 2.950  m/s for a considerable length of time. When younger brother Paul runs a considerable distance, it turns out that he can maintain just this same pace.

A. The brothers decide to a have a race, with Jerome paddling and Paul running, from their farm to their Aunt Annabelle's house and back; their Aunt lives on the riverbank 2020.0  m east of their farm. The bank runs purely east and west along this stretch of the river, and there is a nice path along the bank along which Paul can run. Which brother is going to win, and by how much time?

Give the time difference in seconds. If Paul wins, make the time difference positive. If Jerome wins, make the time difference negative.

Additional info:

The brothers like to visit their Uncle Leo who lives on the southern bank of the river. The river is wide at this point, 1290.0  m across, and their uncle's dock is 225.0  m to the east of the point which is directly across the river from the brothers' house. Paul is not nearly as strong a paddler as is Jerome, but paddling together they can maintain a paddling speed of 3.245  m/s in the farm pond. Jerome knows that if they point their canoe due south, they will always end up to the east of Uncle Leo's dock by the time they have paddled across the river. He wants to know in which direction they should head to arrive exactly at Uncle Leo's dock without any wasted effort. Paul is finally able to determine the proper direction by using the Law of Sines, which he has learned in his high school geometry class. Make a proper drawing to express the sum of velocities for this problem, and figure out how Paul was able to determine the direction.

B. In which direction must they head if they wish to paddle directly to Uncle Leo's dock? Give the direction in terms of an angle measured to the West of the Southerly direction. Also, if they point their canoe in the optimal direction, what will be the speed of their canoe with respect to the bank as they are crossing?

Give your answer as an ordered pair, with the optimal direction in which to point their canoe first, followed by a comma, then followed by the resulting speed of the canoe with respect to the bank. Give the direction in terms of an angle measured to the West of the Southerly direction.

C. If they point their canoe in the optimal direction, how long will it take for them to cross the river?

D. Once they have finished their visit with Uncle Leo, in which direction must the brothers head in order to return directly to their own dock on the north shore of the river? Also, if they point their canoe in the optimal direction for the return trip, then what will be the speed of their canoe with respect to the bank as they are crossing?

Give your answer as an ordered pair, with the optimal direction in which to point their canoe first, followed by a comma, then followed by the resulting speed of their canoe with respect to the bank. Give the direction in terms of an angle measured to the West of the Northerly direction.

E. If they point their canoe in the optimal direction for the return trip, then how long will the return trip take?

Additional info: The brothers have a cousin Dewey who is a pilot, and who runs a crop dusting service. One windy day, Dewey gives the brothers a ride in his crop dusting plane. The brothers observe that they are flying due east, following the river, and by timing their passage over known landmarks along the riverbank, the boys are able to determine that their ground speed is 113.0  km/h . Dewey doesn't have a working compass in the beat-up old plane, but he does have a working air speed indicator, and the boys observe that their air speed is 110.0  km/h . The boys also observe that the plane is not pointed east, but rather somewhat south of east, and the weather report that they hear on Dewey's transistor radio tells them that the windspeed on this day is 21.00  km/h (but the report does not give the wind direction). Cousin Dewey asks the boys to figure out the wind direction. After some effort, the boys finally determine the wind direction by using the Law of Cosines. Draw a proper velocity vector triangle for this case, and figure out how the boys were able to determine the wind direction.

F. What is the wind direction?

Give the direction in terms of an angle measured counterclockwise from East when looking Down; for example, if the direction were Northeast, the angle would be 45 degrees, Northwest would be 135 degrees, Southwest would be 225 degrees, etc.

G. In what direction is the old crop dusting plane headed?

Give the direction in terms of an angle measured to the South of the Easterly direction.

Answer 1

solve 1st 4 July Friday 2010 N subparts July Saturday Paul +ive I am oblige to 2010 o 101-174. warm any problem 100-175. Wk-27 Jerome -live Jerome paul can can paddle run 2.950 m/s 2.950 m/s river nins East at 1.400 m/s. So total time Jerome take = 464.4 + 1303 = 1767 paul wins by 398. I see And A) Part 8 sunce leo} 1290 m wide 225 m East Speed = 3,245 m/s = 1.400m/ let in Brothers head go west of south cose Time taker 2020 2 2.950 Time & Distance speed = 1369.5 sec Time take by Jerome (1st to reach aunt's place ) while going he is assisted by river Time (going, = 2020 (2.950 1.400) z 464.4 sec Leo + So there will be 2 components of their canoe's velocity is directly perpedicular component = ve caso and along the river component = V₂ - Vesino uit assume they take time t see to reach unde leo's, Vecoso t = 1290 (Vy - vesine) t = 225 flow will While returning to farm river reduce his speed Time Creturning) - 2020 (2.950 - 10400) 21303 sec JULY 2010 MTWTFSS 12 S6T 10 11 12 13 14 15 16 17 18 20 21 22 23 36 21 x 29 30 31 AUGUST 2010 MTWT 30 31 1 33567 9 in 1 2 3 14 15 16 17 18 19 20 21 22 23 24 25 26 27 2 29 The very af helving creates strengths of its own the lyst for the people who make the best of the way things tumot

Ve cose - vesine a 1290 225 8 Time (t) = 1290 vuose = 225. Vy-Vesino 3.245 coso = 1.400 - 3.245 Sino 5.733 = te 1290 3. 1304 225 0.5454 412.5 sec Ansc 3.245 Coso = 8.026 - 18.60 sine form let coso be x then sin o=V1-2² Wees ve cose = 1290 Ro Vsino - Vr 3.245 x = 8.026 - 18.60 1-x² Vsino - Vg Leo's Solving 3.245 cose =-8.026 + 18.60 sino similar to part B putting coso= x coso = 0.8186 o = 35.05) 2 x = 0.9647 & coso = x 0 = 15.27 - their relative speed wort to Bank • Velocity I to River = veloso = 3.104 mis velocuity along River - V - Vesino = 0.54 5 4 m/s net 2 vr + Varong - 3.178 m/s relative velocity Ito river = V cos 35.05 = 2,656 mle along the river - Vesino - V = 0.4638 m/s net = √ (2.656) 2 + (0.463 82 2.696 mis ( 35.05, 2.696 m/s Ansd. E) Ans (485.6 sec Ans E) AUGUST 2010 MTWTFSS 30 31 243 0 11 12 13 16 17 19 20 21 JULY 2010 NT WTS 1 2 3 4 5 6 7 8 10 11 13 14 15 16 17 18 19-20 21 22 23 24 25 26 27 28 29 30 31 AnsB | 15.27° 3.178 m/s Li s ten tout blossom and fruit alive Goals are not only necessary to motivate us. They are essential to really keep