A swimmer heads directly across a river, swimming at 1.2 m/s relative to the water. She arrives at a point 48 m downstream from the point directly across the river, which is 77 m wide.

(a) What is the speed of the river current?

(b) What is the swimmer's speed relative to the shore?

(c) In what direction should the swimmer head so as to arrive at the point directly opposite her starting point?

(a) The swimmer arrives at a point 48 m downstream from the point directly across the river, which is 77 m wide. Let's call the speed of the river current "x".

Since the swimmer's speed relative to the water is 1.2 m/s and the river width is 77 m, the time it takes for the swimmer to cross the river is 77 m / 1.2 m/s = 64.17 s.
During this time, the river current pushes the swimmer downstream by 48 m. Therefore, the speed of the river current can be calculated by dividing the distance downstream by the time taken: x = 48 m / 64.17 s = 0.75 m/s.

(b) To find the swimmer's speed relative to the shore, we need to calculate the resultant velocity. The swimmer is heading directly across the river at 1.2 m/s relative to the water, and the river current is pushing her downstream at 0.75 m/s.
Using the Pythagorean theorem, the resultant velocity can be found: sqrt((1.2 m/s)^2 + (0.75 m/s)^2) ≈ 1.43 m/s.
So, the swimmer's speed relative to the shore is approximately 1.43 m/s.

(c) In order for the swimmer to arrive at the point directly opposite her starting point, she needs to steer her path appropriately. Since there is a river current pushing her downstream, she should aim slightly upstream.
To be more specific, she should aim such that her velocity relative to the water has a slight component directed upstream, counteracting the downstream flow of the river. This will allow her to end up directly opposite her starting point.
But hey, if she wants to take the scenic route and see more of the river, she can always ignore my advice. Just don't blame me if she ends up in a different neighborhood!

To find the speed of the river current and the swimmer's speed relative to the shore, we can use the concept of vector addition.

Let's assume the speed of the river current is Vc, and the swimmer's speed relative to the shore is Vs.

(a) To find the speed of the river current, we need to use the relationship between the swimmer's speed relative to the water and the swimmer's speed relative to the shore:

Vs = Vw + Vc

Where Vw is the swimmer's speed relative to the water.
Given that the swimmer's speed relative to the water is 1.2 m/s, and she arrived 48 m downstream from the point directly across the river, we can use the formula:

48 m = Vw * t

Where t is the time taken to cross the river. Since the swimmer heads directly across the river, the time taken to cross the river is the same as the time taken to travel downstream, which is given by:

t = 48 m / Vs

Substituting the value of Vw:

48 m = 1.2 m/s * (48 m / Vs)
48 m = 57.6 m^2 / Vs
48 m * Vs = 57.6 m^2
Vs = 57.6 m^2 / 48 m
Vs = 1.2 m/s

Now we can find the speed of the river current:

Vc = Vs - Vw
Vc = 1.2 m/s - 1.2 m/s
Vc = 0 m/s

Therefore, the speed of the river current is 0 m/s.

(b) The swimmer's speed relative to the shore is Vs, which we have already found to be 1.2 m/s.

(c) To find the direction the swimmer should head to arrive at the point directly opposite her starting point, we need to consider the cross-stream displacement and the downstream displacement.

The cross-stream displacement is the distance directly across the river, which is given as 77 m. The downstream displacement is the distance the swimmer has traveled downstream, which is given as 48 m.

To reach the point directly opposite her starting point, the swimmer needs to combine these two displacements using vector addition. The swimmer should therefore head at an angle opposite to the direction of the river current.

Since we found earlier that the river current has a speed of 0 m/s, the swimmer should head directly perpendicular to the river, which is directly opposite her starting point.

Therefore, the swimmer should head straight across the river to arrive at the point directly opposite her starting point.

To solve this problem, we can break it down into three parts: finding the speed of the river current, finding the swimmer's speed relative to the shore, and determining the direction the swimmer should head to arrive at the point directly opposite her starting point.

(a) To find the speed of the river current, we need to use the concept of relative velocity. The swimmer's velocity relative to the shore (V_s) is the vector sum of her velocity relative to the water (V_w) and the velocity of the river current (V_c):

V_s = V_w + V_c

In this case, we know the swimmer's velocity relative to the water (V_w) is 1.2 m/s (given), and we need to find V_c. Since the swimmer is swimming directly across the river, her velocity relative to the shore is perpendicular to the river current. Therefore, the horizontal components of the velocities must be equal:

V_s_x = V_w_x + V_c_x
0 = 1.2 m/s + V_c_x

We also know that the swimmer arrives at a point 48 m downstream from the point directly across the river, which means her displacement in the y-direction (downstream) is -48 m. We can use this information to set up another equation:

V_s_y = V_w_y + V_c_y
-48 m = 0 + V_c_y

Since V_c_y = 0 (the river current does not contribute to the swimmer's displacement in the y-direction), we can rewrite the equation as:

V_s_y = 0

From this equation, we can determine that the swimmer's speed in the y-direction (downstream) is 0 m/s.

Now, let's go back to the first equation. Since V_w_x = V_w, we can rewrite it as:

0 = 1.2 m/s + V_c_x

Solving for V_c_x, we find that V_c_x = -1.2 m/s.

We can conclude that the speed of the river current is 1.2 m/s to the left.

(b) The swimmer's speed relative to the shore (V_s) is the magnitude of her velocity relative to the shore:

|V_s| = sqrt(V_s_x^2 + V_s_y^2)

From our previous calculations, we know V_s_x = 1.2 m/s and V_s_y = 0 m/s. Plugging these values into the equation, we have:

|V_s| = sqrt((1.2 m/s)^2 + (0 m/s)^2)
|V_s| = sqrt(1.44 m^2/s^2 + 0 m^2/s^2)
|V_s| = sqrt(1.44 m^2/s^2)
|V_s| = 1.2 m/s

Therefore, the swimmer's speed relative to the shore is 1.2 m/s.

(c) To determine the direction the swimmer should head, we need to find the angle between her velocity relative to the shore and the x-axis. This angle is equal to the inverse tangent of V_s_y/V_s_x:

θ = arctan(V_s_y/V_s_x)

In this case, V_s_x = 1.2 m/s and V_s_y = 0 m/s, so the equation becomes:

θ = arctan(0/1.2)

Since the y-component of the swimmer's velocity is 0, we can conclude that she should head directly across the river (perpendicular to the x-axis) in order to arrive at the point directly opposite her starting point.

Therefore, the swimmer should head straight across the river to reach the desired destination.

a. Tan A = 48/77 = 0.62338

A = 31.94o

Tan31.94 = Vc/1.2
Vc = 1.2*Tan31.94 = 0.748 m/s = Speed of
the current.

b. Vs = 1.2/Cos31.94 = 1.414 m/s = Speed of swimmer.

c. The current shifted the swimmer 31.94o E. of the desired point. To offset the affect of the current, the
swimmer must head 31.94o west of the
desired point.