A motorboat heads due west at 10 m/s. The river has a current that travels south at 6 m/s.

a) what is the resultant velocity?

b) If the river is 200 m wide, how long doe sit take the boat to cross the river?

c) How far downstream is the boat when it reaches the other side?

Please, please, please help me and explain.
I have a test on all kinds of problems like this tomorrow morning.

b)

I am going to do b first because it is easy
I assume the river is North to South so west is straight across.
Our velocity component straight across is 10m/s
so time to cross = 200m / 10m/s = 20 seconds

Now c) because we know how long we went at 6 m/s downstream
6*20 = 120 meters downstream

Now finally a, the long one
magnitude of resultant velocity = sqrt (10^2 + 6^2) = sqrt (136) = 11.662 m/s
tan angle below straight across = 6/10
so angle south of west = 30.964 degrees

Okay so when we were learning this, we had to add the components and do sine and cosine. Do i have to do that in this problem? If so, where?

Well, I used tangent instead of sine and cosine but if you wanted to use sine or cos do this

We have the hypotenuse = 11.662
so sine angle = 6/11.662
so angle = sin^-1 .5145
which of course is 30.964 again
or we could use
cosine angle = 10/11.662 = .8575
so angle = cos^-1 .8575
which lo and behold is 30.963 close enough :)

You probably gave me a problem that was too easy because you were already given the west and south components instead of a direction and a speed.

Ah got it. Thank you. So another kind of problem that is confusing me, probably because i'm making it harder than it is, is when forces are acting on a point:

Three forves act on a point concurrently. Forve a is 80N at 220 degrees, force b is 70N at 90 degrees, and forve c is 70N at 0 degrees. Find:

a) resultant force

b) equilibrant force

a) To find the resultant velocity, we need to use vector addition. Since the motorboat is moving due west and the current is moving due south, we can consider their velocities as vectors in a coordinate plane.

Let's label the westward velocity of the motorboat as -10 m/s along the x-axis (negative because it is in the opposite direction of positive x-axis) and the southward velocity of the current as -6 m/s along the y-axis (again negative because it is in the opposite direction of positive y-axis).

To find the resultant velocity, we can use the Pythagorean theorem:

Resultant velocity = √((-10)^2 + (-6)^2) m/s
Resultant velocity = √(100 + 36) m/s
Resultant velocity = √136 m/s

So, the resultant velocity of the boat is approximately 11.66 m/s at an angle of approximately 32.1° south of west.

b) To find the time it takes the boat to cross the river, we can use the formula:

Time = Distance / Velocity

The distance to cross the river is given as 200 m and the velocity we calculated in part a) is approximately 11.66 m/s.

Time = 200 m / 11.66 m/s
Time ≈ 17.15 s

So, it takes approximately 17.15 seconds for the boat to cross the river.

c) To find how far downstream the boat is when it reaches the other side, we can use the formula:

Distance = Velocity × Time

The downstream velocity is the southward velocity of the current, which is -6 m/s. The time to cross the river we calculated in part b) is approximately 17.15 s.

Distance = -6 m/s × 17.15 s
Distance ≈ -102.9 m

Since the downstream distance is negative, it means the boat ends up 102.9 meters upstream on the opposite side.

To summarize:
a) The resultant velocity of the boat is approximately 11.66 m/s at an angle of approximately 32.1° south of west.
b) It takes approximately 17.15 seconds for the boat to cross the river.
c) The boat ends up approximately 102.9 meters upstream on the opposite side.