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
-----------------
I know we're dealing with motion in 2D, but I don't even know which equations to use or how to start. Any help or jumpstarts will be greatly appreciated.
To solve this problem, we can use the concepts of relative velocity and vector addition.
a) To minimize the time spent in the water, you should head in the direction that reduces the effective velocity across the river. This means swimming perpendicular to the river current. The angle between your swimming direction and the direction of the stream is 90°.
b) To find how far downstream you will be carried, we need to calculate the displacement across the river. Since the water is flowing uniformly, the displacement should be proportional to the time spent in the water.
Let's consider the motion in the x-direction (across the river). The speed of the water is 2.53 m/s, and your swimming speed is 1.74 m/s.
The time taken to cross the river, t, can be calculated using the equation:
t = distance / speed
The distance is given by the width of the river, which is 69.8 m.
The speed in the x-direction, which is the difference between the water velocity and your swimming velocity, is:
speed_x = water speed - swim speed = 2.53 m/s - 1.74 m/s = 0.79 m/s
Plugging in these values, we can find the time t:
t = 69.8 m / 0.79 m/s ≈ 88.35 s
Since the displacement in the x-direction is equal to the speed in the x-direction multiplied by time, we have:
displacement_x = speed_x * t = 0.79 m/s * 88.35 s ≈ 69.78 m
Therefore, you'll be carried approximately 69.78 m downstream.
c) To minimize the distance downstream that the river carries you, you should head directly across the river along the direction of the stream.
d) In this case, since you're swimming directly across the river, the downstream distance you'll be carried is zero.
To summarize:
(a) You should head 90° from the direction of the stream.
(b) You'll be carried approximately 69.78 m downstream.
(c) You should head directly along the direction of the stream.
(d) You won't be carried downstream if you swim directly across the river.
maybe go with the current to cross faster and upstream to make a shorter distance
I'm only in gr.9 but I hope this helps