Two girls are at the same point on one side of a river that is 40m wide with a current flowing at 1.00m/s. Simultaneously, the dive into the water in an attempt to swim to the other bank of the river directly across from where they started. Both swim at 2m/s relative to the water, but Girl A directs herself so that her net motion takes her straight across while Girl B keeps her body perpendivular to the current and consequently lands downstream. After landing she runs along the shore to where Girl A landed at a speed of 6 m/s. Which girl arrives first and by how much?
Girl Two, is swimming up stream at an angle of arcsin(1/2) or 30 degrees. So she takes time 40/2*.866seconds to get across, or 23.09 seconds to get across. check that math.
Girl one, takes 20 seconds to get across, but then it takes her distance downstream at running. Distance downstream=20sec*1m/s=20m
total time= 20/6+20=23.3 seconds
So you better fine tune my calcs.
^Thanks
To determine which girl arrives first and by how much, we need to calculate the time it takes for each girl to reach the other side of the river. Let's break down the steps to find the solution:
Step 1: Calculate the time it takes for Girl A to swim straight across the river.
Since Girl A swims directly across the river, her speed relative to the water is the same as her speed relative to the shore, which is 2 m/s. The distance she needs to swim is the width of the river, which is 40 m. Therefore, the time taken by Girl A to cross the river is:
Time_A = distance / speed = 40 m / 2 m/s = 20 s
Step 2: Calculate the time it takes for Girl B to reach the other side of the river.
Girl B swims perpendicular to the current. We can consider her speed relative to the water as the vector sum of her swimming speed and the speed of the current. Therefore, her speed relative to the water is:
Speed_B = 2 m/s + 1 m/s = 3 m/s
Using this speed and the same distance of 40 m, we can calculate the time taken by Girl B to cross the river:
Time_B = distance / speed = 40 m / 3 m/s ≈ 13.33 s
Step 3: Calculate the time it takes for Girl B to run along the shore.
After landing downstream, Girl B needs to run along the shore to where Girl A landed. The distance between the two landing points is 40 m. The speed at which Girl B runs is 6 m/s. So, the time taken by Girl B to run along the shore is:
Time_run = distance / speed = 40 m / 6 m/s ≈ 6.67 s
Step 4: Compare the total times taken by both girls.
The total time taken by Girl A is the time to swim directly across (20 s).
The total time taken by Girl B is the time to swim across (13.33 s) plus the time to run along the shore (6.67 s).
Therefore, Girl A arrives first in 20 seconds, while Girl B arrives in a total time of 20 s (13.33 s + 6.67 s = 20 s). Both girls arrive at the same time, with no time difference.