Relative Velocities Jerome and Paul are competitive brothers. They live on a sma
ID: 1838879 • Letter: R
Question
Relative Velocities
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 1.650 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 3.150 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.
1) 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?
485 s is the answer for this one.
The brothers like to visit their Uncle Leo who lives on the southern bank of the river. The river is wide at this point, 1310.0 m across, and their uncle's dock is 155.0 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 3.780 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.
2) 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?
Optimal direction , speed with respect to the bank = degrees West of North, m/s
3) If they point their canoe in the optimal direction for the return trip, then how long will the return trip take?
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 117.0 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 111.0 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 18.00 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.
4) What is the wind direction? degrees counterclockwise from East
5) In what direction is the old crop dusting plane headed? degrees South of East
Explanation / Answer
Ans 1.
Step1:Picturizing theproblem
(i) Paul's speed is constant during onward and return jouney.
(ii) The river affects Jerome's speed.
(iii) Total distance= 2X2020m
Step 2: Jerome's speed & time taken
Speed of Jerome during onward journey = own speed(3.150 m/s) + rivers speed (1.650m/s0=4.800m/s
Speed of Jerome while returning= own speed -river's speed=1.500 m/s
So time taken by jerome during onward journey = 2020m/ (4.800m/s)=420.83 s
Time taken by Jerome while returning= 2020m/(1.500m/s)=1346.67
Hence total time taken by Jerome T1=1767.50 s
Step3: Paul's speed & time
Paul's speed is =3.150m/s
Total distance he covers=2X2020=4040m
Hence time taken=4040m/(3.150m/s) T2=1282.54 s
Step4: The answer
As T2<T1, Paul wins.
Margin is T1-T2=1787.5-1282.54=485 s