You are riding on a jet ski directed at an angle upstream on a river flowing with a speed of 2.8 m/s. If your velocity relative to the ground is 8.2 m/s at an angle of 25.0° upstream, what is the speed of the jet ski relative to the water?
To find the speed of the jet ski relative to the water, we can use vector addition.
Let's break down the given information:
- The river is flowing with a speed of 2.8 m/s.
- Your velocity relative to the ground is 8.2 m/s at an angle of 25.0° upstream.
We need to find the speed of the jet ski relative to the water, which means we need to determine the magnitude of the velocity vector of the jet ski in the water.
First, let's resolve the velocity vectors into their components:
The velocity of the river can be split into horizontal and vertical components. Since the river is flowing upstream, the vertical component is positive and the horizontal component is zero.
Vertical component of river velocity (Vry) = 2.8 m/s * sin(0°) = 0 m/s
Horizontal component of river velocity (Vrx) = 2.8 m/s * cos(0°) = 2.8 m/s
The velocity of the jet ski relative to the ground can also be split into horizontal and vertical components:
Vertical component of jet ski velocity relative to the ground (Vgy) = 8.2 m/s * sin(25.0°)
Horizontal component of jet ski velocity relative to the ground (Vgx) = 8.2 m/s * cos(25.0°)
Since the velocity of the jet ski relative to the water is the difference between the velocity of the jet ski relative to the ground and the velocity of the river, we can use vector subtraction:
Vertical component of jet ski velocity relative to the water (Vwy) = Vgy - Vry
Horizontal component of jet ski velocity relative to the water (Vwx) = Vgx - Vrx
Finally, we can calculate the magnitude of the velocity vector of the jet ski in the water (Vw) using the Pythagorean theorem:
Vw = sqrt((Vwx)^2 + (Vwy)^2)
Now, plug in the values and calculate:
Vertical component of jet ski velocity relative to the water (Vwy) = 8.2 m/s * sin(25.0°) = 3.46 m/s
Horizontal component of jet ski velocity relative to the water (Vwx) = 8.2 m/s * cos(25.0°) = 7.42 m/s
Magnitude of the velocity vector of the jet ski in the water (Vw) = sqrt((7.42 m/s)^2 + (3.46 m/s)^2)
Therefore, the speed of the jet ski relative to the water is approximately 8.07 m/s.