The water in a river flows uniformly at a constant speed of 2.53 m/s between parallel banks 69.8 m apart. You are to deliver a package directly across the river, but you can swim only at 1.74 m/s.

(a) If you choose to minimize the time you spend in the water, in what direction should you head? ______° from the direction of the stream

(b) How far downstream will you be carried? ______m

(c) If you choose to minimize the distance downstream that the river carries you, in what direction should you head? ______° from the direction of the stream

(d) How far downstream will you be carried? ______m

Any help as to how to start this or which formula to apply would be great. I'm really confused with motion in two dimensions.

To solve this problem, we'll need to break it down into two components: the motion parallel to the river's flow (downstream) and the motion perpendicular to the river's flow (across the river).

Let's start with part (a):

(a) To minimize the time spent in the water, you need to head directly across the river. This means you need to swim perpendicular to the direction of the river's flow. To determine the angle, we can use the fact that the speed of your swim relative to the river's flow is given by the Pythagorean theorem:

speed_relative = √(speed_swim^2 + speed_river^2)

Where:
speed_swim = 1.74 m/s (your swimming speed)
speed_river = 2.53 m/s (the speed of the river's flow)

Plugging in the values, we can calculate the speed_relative:

speed_relative = √(1.74^2 + 2.53^2)
speed_relative ≈ 3.07 m/s

Since you want to minimize the time spent in the water, you should swim directly across the river, which means perpendicular to the river's flow. Therefore, the direction you should head is 90° from the direction of the stream.

Now let's move on to part (b):

(b) To find out how far downstream you will be carried, we need to determine the time it takes for you to cross the river. Since we are assuming that the river flows uniformly, we can use the equation:

time = distance / speed_relative

Where:
distance = 69.8 m (the width of the river)
speed_relative ≈ 3.07 m/s (as calculated in part (a))

Plugging in the values, we can calculate the time it takes to cross the river:

time = 69.8 m / 3.07 m/s
time ≈ 22.76 s

Since the river flows at a speed of 2.53 m/s, you will be carried downstream during this time. To find out the distance downstream, we can use the equation:

distance_downstream = speed_river * time

Where:
speed_river = 2.53 m/s (the speed of the river's flow)
time ≈ 22.76 s (as calculated above)

Plugging in the values, we can calculate the distance downstream:

distance_downstream = 2.53 m/s * 22.76 s
distance_downstream ≈ 57.35 m

Therefore, if you swim directly across the river, you will be carried approximately 57.35 meters downstream.

Now, let's move on to part (c):

(c) To minimize the distance downstream that the river carries you, you need to head directly downstream, parallel to the river's flow. This means you should swim in the same direction as the river. Again, we will use the fact that the speed of your swim relative to the river's flow is given by the Pythagorean theorem:

speed_relative = √(speed_swim^2 + speed_river^2)

Plugging in the values, we can calculate the speed_relative:

speed_relative = √(1.74^2 + 2.53^2)
speed_relative ≈ 3.07 m/s

Since you want to minimize the distance downstream, you should swim in the same direction as the river's flow. Therefore, the direction you should head is 0° from the direction of the stream.

Finally, let's move on to part (d):

(d) If you swim directly downstream, you won't be carried across the river. Therefore, the distance downstream you will be carried in this case will be 0 meters.

To summarize:

(a) You should head perpendicular to the direction of the stream, at 90° from the direction of the stream.
(b) You will be carried approximately 57.35 meters downstream.
(c) You should head in the same direction as the stream, at 0° from the direction of the stream.
(d) You will be carried 0 meters downstream.

I hope this helps! Let me know if you have any further questions.

To solve this problem, we can break it down into components and use some basic trigonometry.

(a) To minimize the time spent in the water, you should head in the direction opposite to the flow of the river. This means you need to subtract the velocity of the river from your swimming velocity to get the direction.

(b) To find how far downstream you will be carried, we can use the formula:

distance = velocity × time.

Since the time you will spend in the water is the same as the time it takes for you to swim across the river, we can use the distance formula:

distance = velocity × time.

The distance across the river is given as 69.8 m, and your swimming speed is 1.74 m/s. Therefore, the distance downstream you will be carried is:

distance = (2.53 m/s - 1.74 m/s) × time

Substituting the given values:

distance = (0.79 m/s) × time.

(c) To minimize the distance downstream you will be carried, you should head exactly perpendicular to the flow of the river. This means choosing a direction that is 90° from the direction of the stream.

(d) The distance downstream you will be carried can be calculated using the same distance formula as before. However, since you are heading directly perpendicular to the flow of the river, your swimming velocity and the downstream velocity are no longer in the same direction. To find the distance downstream, you will need to consider the vector components.

The component of the river flow velocity in the downstream direction will be:

downstream_velocity = river_velocity × (cosine of the angle between the stream direction and downstream direction).

Since the angle between the stream direction and downstream direction is 90°, the cosine of 90° is 0. Therefore, the downstream_velocity component will be 0.

The distance downstream you will be carried is given as:

distance = downstream_velocity × time.

Since the downstream velocity component is 0, the distance downstream will also be 0.

I suppose you have done vectors to do this exercise.

Basically, you'd need to represent each velocity in magnitude and direction, and proceed to add them up verctorially to get the resultant.

For the first part, to swim across in minimum time, I would aim right across the river (90° to the flow), and let the river take me wherever I end up. Any other direction will cost me a component to fight the current or help the current. The time is exactly the distance divided by the swimmer's still water speed.

The velocities can be represented by a triangle ABC, in which AB(1.74) is the swimmer's still water velocity across the river. BC(2.53) is the river's current along the river towards downstream. ∠ABC is a right angle. AC represents the resultant velocity and can be obtained by Pythagoras theorem. (not required for answer).

For parts (C) and (D), it is a little more tricky because the swimmer's speed is slower than the current's. He will not be able to make it straight across, but he can minimize the distance carried downstream by making the angle BCA as big as possible. I will leave it to you to find the solution.