Relative Velocities with angles

[Jtappan]

Homework Statement

A dolphin wants to swim directly back to its home bay, which is 0.79 km due west. It can swim at a speed of 4.14 m/s relative to the water, but a uniform water current flows with speed 2.93 m/s in the southeast direction.

(a) What direction should the dolphin head?
_______° N of W
(b) How long does it take the dolphin to swim the 0.79-km distance home?
________min

The Attempt at a Solution

First I have been trying to find all of my variables in both directions. The angle should be 45 degrees for the 2.93m/s in the south east directions, I am pretty sure. Then I am just completely lost after that. If it is a 45 degree angle then you do 2.93sin45 and get 2.07 roughly. This is where it loses me because there are two different velocities for the Dolphin relative to the water.

 

[learningphysics]

Let theta be the angle of the dolphin's velocity from the positive east axis...

So the dolphin goes 4.41cos(theta) in the horizontal direction...

He goes 4.41sin(theta) in the vertical direction.

What is the horizontal and vertical component of the water...

What can you say about the sum of the vertical component of the water + the vertical component of the dolphin's velocity? That should let you get theta... also note, theta should be bigger than 90 degrees... because the dolphin needs to go west...

 

[m2003]

The dolphin is swimming at 4.14m/s and the water is moving at 2.07m/s in the opposite direction. So I am not sure how to combine those two velocities to find the actual velocity of the dolphin relative to the ground. I would approach this problem by breaking it down into its components and using vector addition to find the solution.

First, we need to find the velocity of the dolphin relative to the ground. This can be done by using vector addition, where we add the velocity of the dolphin relative to the water (4.14 m/s) with the velocity of the water current (2.07 m/s in the opposite direction). This results in a velocity of 2.07 m/s for the dolphin relative to the ground.

Next, we need to find the direction that the dolphin should head in order to swim directly back to its home bay. This can be done by using trigonometry. Since the dolphin wants to swim due west, we can use the tangent function to find the angle. The tangent of an angle is equal to the opposite side (2.07 m/s) divided by the adjacent side (4.14 m/s). Therefore, the angle that the dolphin should head is 26.6 degrees north of west.

To find the time it takes for the dolphin to swim the 0.79 km distance home, we can use the formula time = distance/velocity. Plugging in the values, we get time = 0.79 km / 2.07 m/s = 0.38 minutes or roughly 23 seconds.

In conclusion, the dolphin should head in a direction 26.6 degrees north of west and it will take approximately 23 seconds for it to swim the 0.79 km distance home.