Vector Math Problem (Boat crossing a River)

[zetlearn]

Homework Statement

(a) At what angle Sumon has to drive the boat to reach college at time?

Context: The 4 km width river is situated in front of Sumon's house and his college is in opposite of that river. One day Sumon started his journey for college at 7.30 am by making an angle of 50° with a velocity of 5 km/h. College starts at 8.30. Velocity of current is 2 km/h.

Relevant Equations

The 4 km width river is situated in front of Sumon's house and his college is in opposite of that river. One day Sumon started his journey for college at 7.30 am by making an angle of 50° with a velocity of 5 km/h. College starts at 8.30. Velocity of current is 2 km/h.

My answer is 130.54...
Is that correct?

 

[berkeman]

Welcome to PF.

Please describe how you calculated that answer, and show all of your work that led to it. Thanks.

Oh, and 130 degrees with respect to what?

 

[zetlearn]

Welcome to PF.

Please describe how you calculated that answer, and show all of your work that led to it. Thanks.

Oh, and 130 degrees with respect to what?

Respect to Direction of Current

 

[zetlearn]

 

[zetlearn]

Any idea how to solve it please??

 

[berkeman]

Sorry, several things don't make sense. The problem statement seems to be giving you an angle of travel (with respect to a line that goes straight across the river, not pointing in the direction of the current). They give you a start time and the velocities, so it seems like they want to know when the boat reaches the other side (and probably want you to verify that the boat makes it straight across). So why are you trying to calculate an angle? Aren't you supposed to calculate the travel time?

 

[zetlearn]

Sorry, several things don't make sense. The problem statement seems to be giving you an angle of travel (with respect to a line that goes straight across the river, not pointing in the direction of the current). They give you a start time and the velocities, so it seems like they want to know when the boat reaches the other side (and probably want you to verify that the boat makes it straight across). So why are you trying to calculate an angle? Aren't you supposed to calculate the travel time?

The question is: (a) At what angle Sumon has to drive the boat to reach college at time?

 

[nasu]

Then whay do you say that it travels at 50 degrees?
The text says: "by making an angle of 50°" .
You need a good, coherent question before even thinking of what the answer may be.

 

[jbriggs444]

The question is: (a) At what angle Sumon has to drive the boat to reach college at time?

You seem to have interpreted the question as:

"At what angle (relative to directly downstream) must Sumon drive so that he covers exactly four miles (not necessarily in the right direction) in one hour of travel time?"

You correctly calculate that after one hour, the river will have carried Sumon 2 miles downstream. The boat will have travelled 5 miles relative to the water. We want the vector sum of that 2 mile vector and the 5 mile vector to have a magnitude of 4 miles. So you use the law of cosines and solve for the angle between the 2 mile vector (directly downstream) and the 5 mile vector (the direction taken by the boat).

Let us take this to the next step. Where will Sumon be after one hour if he follows your 130 degree heading?

Let us do it by components. We will use the $x$ axis for the direction of the current flow and the $y$ axis for the direction across the river toward the college. We will use $w$ for the distance travelled by the water and $b$ for the distance travelled by the boat relative to the water. We will use $s$ for the total distance.

$w_x = 2$, $w_y = 0$
$b_x = 5 \cos 130 = -3.2$, $b_y = 5 \sin 130 = 3.8$
$s_x = -1.2$, $s_y = 3.8$

So he has made it only 3.8 miles across the river and has ended up 1.2 miles upstream.

Maybe he needs to aim more nearly directly across so that he actually hits the college instead of missing? Is it really a problem if he arrives early?