Math and physics problem

Question

Two girls are at point x on one side of a river, 50.0 m wide and having a current of 1.0 m/s, flowing as shown. Simultaneously, they drive into the water in an attempt to reach point Y, directly opposite X. Both swim at 2.0 m/s relative to the water, but one directs herself so that her net motion corresponds to XY, while the other keeps her body perpendicular to the current and consequently lands at point Z. After landing, she runs along the shore to point Y at a speed of 6.0 m/s. Which girl arrives first, and by how much time does she beat the other?

Answer 1

I can’t see the image so I am giving my best guess on what it looks like.

Girl 1 - directs her velocity 2m/s straight toward point Y while the current pushes sideways 1m/s so she lands at point Z

Girl 2 - directs her velocity 2m/s at a diagonal opposite of the current to cancel any sideways motion and land straight at Y

To find how much time it takes them to reach the other side, we need to find their straight components. Then we can do

speed = distance / time

using the width of the river to find the time to cross the river. Then for girl 1 we will figure out how far sideways she went to determine how much extra she spent running from Z to Y.

1. determine the components of each velocity

Girl 1 -

straight component = 2m/s

sideways component = 0 m/s given by the girl so she will move with a sideways component of the current = 1m/s

Girl 2-

diagonal (hypotenuse) = 2 m/s

sideways component = 1 m/s opposite of current so she will cancel the sideways component = 0m/s

straight component = Pythagorean theorem a^2+b^2=c^2

1^2 + b^2 = 2^2

1 + b^2 = 4

b^2 = 3

b = sqrt 3 = 1.73 m/s straight component

1. use these straight components to figure out the time it took to cross the water

speed = distance / time

speed = straight component

distance = width of river = 50m

Girl 1:

2 = 50/t multiply both sides by t

2t = 50 divide by 2

t = 25.0 seconds to cross the water

Girl 2:

1.73 = 50/t

1.73t = 50

t = 28.9 seconds to cross the water

1. Now, for girl 1 we have to figure out how far sideways she went and determine the time it takes to run back to Y

speed = distance / time

We know she is in the water for 25 seconds and moving sideways with the speed of the current 1 m/s. We are looking for the sideways distance

1 = distance / 25 multiply by 25

distance = 25 m

Girl 1 moves sideways by 25m —> this is the distance between Y and Z

1. determine the time it takes to run from Z to Y

speed = distance / time

The speed of her run is 6m/s and the distance is 25m.

6 = 25/t multiply by t

6t = 25 divide by 6

t = 4.2 seconds

We need to add this time onto the time it took Girl 1 to cross the water to find her total time

Girl 1 total time = 25+4.2 = 29.2 seconds

Girl 2 total time = 28.9 seconds

Girl 2 beats Girl 1

time difference = 29.2 - 28.9 = 0.3 seconds

The girl who directed herself against the current and landed straight across at Y wins. :)