How Does Current Affect a Swimmer's Path Across a River?

[Draggu]

Homework Statement

A swimmer can swim at a speed of 1.80m/s in still water. If the current in a river 200m wide is 1.0m/s [E], and the swimmer starts on the south bank and swims so that she is always headed directly across the river, determine:

a) The swimmer's resultant velocity, relative to the river bank.
b) How long she will take to reach the far shore.
c) How far downstream she will land (from the point opposite her starting point)

Homework Equations

The Attempt at a Solution

a) Draw a vector 1.8m/s north, and then 1.0m/s east?

If so,

2.06m/s[N29$$\circ$$E]

b) t = d/v
t=(200m)/(1.8m/s)
=111s

c) d=v*t
d=(1m/s)(111s)
=111m[E]

 

[anathema]

Thank you for your inquiry. I would like to provide you with a more detailed explanation and solution to the problem you have presented.

a) To determine the swimmer's resultant velocity, we need to take into account the velocity of the swimmer and the velocity of the current. Since the swimmer is always headed directly across the river, we can use vector addition to find the resultant velocity. This can be done by drawing a vector representing the swimmer's velocity of 1.8m/s north and another vector representing the current's velocity of 1.0m/s east. Using the Pythagorean theorem, we can calculate the magnitude of the resultant velocity:

Resultant velocity = √(1.8^2 + 1.0^2) = √3.24 = 1.8m/s

To determine the direction of the resultant velocity, we can use trigonometric functions to find the angle between the resultant velocity and the north direction:

tanθ = (1.0/1.8)
θ = tan^-1(1.0/1.8) = 29.74°

Therefore, the swimmer's resultant velocity, relative to the river bank, is 1.8m/s at an angle of 29.74° east of north.

b) To find the time it takes for the swimmer to reach the far shore, we can use the formula t = d/v, where d is the distance and v is the velocity. In this case, the distance is 200m (width of the river) and the velocity is the magnitude of the resultant velocity, which we calculated to be 1.8m/s. Therefore:

t = (200m)/(1.8m/s) = 111.11s

c) To determine how far downstream the swimmer will land, we can use the formula d = v*t, where v is the velocity and t is the time. In this case, the velocity is the current's velocity of 1.0m/s and the time is the same as the time it takes for the swimmer to reach the far shore, which we calculated to be 111.11s. Therefore:

d = (1m/s)*(111.11s) = 111.11m

This means that the swimmer will land 111.11m downstream from the point opposite her starting point