SOLUTION: Pascal and his friend Fermat live on opposite sides of a river that is 1 km wide. Fermat lives 2 km downstream from Pascal on the opposite side of the river. Pascal can swim at a r
Algebra ->
Vectors
-> SOLUTION: Pascal and his friend Fermat live on opposite sides of a river that is 1 km wide. Fermat lives 2 km downstream from Pascal on the opposite side of the river. Pascal can swim at a r
Log On
Question 1201561: Pascal and his friend Fermat live on opposite sides of a river that is 1 km wide. Fermat lives 2 km downstream from Pascal on the opposite side of the river. Pascal can swim at a rate of 3 km/h and the river’s current has a speed of 4 km/h. Pascal swims from his cottage directly across the river.
a) What is Pascal’s resultant velocity?
b) How far away from Fermat’s cottage will Pascal be when he reaches the other side?
c) How long will it take Pascal to reach the other side?
please help me with this vectors question. it's from my "vectors as veolcity" lesson. thank you so much. Found 2 solutions by math_tutor2020, ikleyn:Answer by math_tutor2020(3816) (Show Source):
You can put this solution on YOUR website!
Answers:
(a) 5 km/hr
(b) 0.67 km approximately (equal to 2/3 km exactly)
(c) 20 minutes (aka 1/3 of an hour)
Explanation:
Here is one way to draw the diagram.
P = Pascal's house
F = Fermat's house
A = Pascal's landing spot if there wasn't a moving current
B = Pascal's landing spot due to river current
kph = km/hr = kilometers per hour
If there wasn't a current, then Pascal swims at 3 kph directly north to reach point A.
Let's find out how long it takes to do so
distance = rate*time
1 km = (3 kph)*(x hrs)
x = 1/3 of an hour
x = 20 minutes
Pascal would take 20 minutes go from P to A if there was no current.
This is the answer to part (c).
But of course there is a river current. The river is moving to the right at 4 kph to push Pascal to land on point B instead of A.
Notice in the diagram above we have a 3-4-5 right triangle.
Use the pythagorean theorem to plug in a = 3 and b = 4, and you should get c = 5.
Therefore, the answer to part (a) is 5 km/hr.
Pascal wants to go 3 kph north, but the river wants to push him 4 kph east
The two have a tug-of-war so to speak, and the winning compromise is along vector PB and the speed is 5 kph. This is the resultant speed. Pascal is going 5 kph along segment PB due to the current pushing him east.
Recall that Pascal took 20 minutes to go from P to A when there wasn't a current.
With the current, Pascal will also take 20 minutes to go from P to B. The vertical component of the velocity remains at 3kph. The only thing different here is the added horizontal component to drift eastward.
This is why the time durations of "from P to A" and "from P to B" are the same.
Focus on the 3-4-5 right triangle.
Let's find the distance from A to B.
distance = rate*time
distance = (4 kph)*(1/3 of an hr)
distance = 4/3 of a km
The distance from A to B is 4/3 km.
The distance from A to F is 2 km since Fermat lives 2 km downstream of Pascal.
AB+BF = AF
(4/3 km) + (BF) = (2 km)
BF = 2 - (4/3)
BF = (6/3) - (4/3)
BF = (6-4)/3
BF = 2/3
BF = 0.67 km approximately
This is the extra distance on land Pascal needs to travel when going from B to F.
This is the answer to part (b).
You can put this solution on YOUR website! .
Pascal and his friend Fermat live on opposite sides of a river that is 1 km wide.
Fermat lives 2 km downstream from Pascal on the opposite side of the river.
Pascal can swim at a rate of 3 km/h and the river’s current has a speed of 4 km/h.
Pascal swims from his cottage directly across the river.
a) What is Pascal’s resultant velocity?
b) How far away from Fermat’s cottage will Pascal be when he reaches the other side?
c) How long will it take Pascal to reach the other side?
~~~~~~~~~~~~~~~~~~~
Pascal participates in two movements, simultaneously.
He swims at the rate 3 km/h in the direction perpendicular to the banks of the river
(and perpendicular to the river' stream).
At the same time, he moves at the rate of 4 km/h along the river, together with the stream
and the current.
The vector of the combined speed is the hypotenuse of a right angle triangle with the legs
of 3 km/h and 4 km/h.
So, the magnitude of his combined rate is v = = = = 5 km/h.
It is the answer to question (a).
To answer question (c), notice that Pascal will cross the river in
= = of an hour = 20 minutes.
It is the answer to question (c).
At that time, the river stream will drift Pascal the distance
4 km/h * (1/3) hour = 4/3 km = 1 km.
Hence, Pascal will reach the opposite bank of the river at the distance 4 - 1 = 2 km from the Fermat's house.
It is the answer to question (b).
Solved. // All questions are answered.
----------------
Notice that I should change the order of my answers comparing with the order of questions,
in order for the solution could follow the normal/regular human logic.
(