A river is flowing 4.0 m/s to the east. A boater on the south shore plans to reach a dock on the north shore 30.0 Degrees downriver by heading directly across the river. What should be the boat's speed relative to the water?
Sin30=4/V
solve for V
To find the boat's speed relative to the water, we can use the concept of vector addition. Relative speed is the difference between the boat's speed and the velocity of the river.
Let's break down the given information into components:
The river's velocity is given as 4.0 m/s to the east. As the boat is perpendicular to the river's flow, the river's velocity won't affect the boat's northward motion. So, we only need to consider the eastward velocity component from the river, which is 4.0 m/s.
The desired downriver direction is 30.0 degrees downriver. We can break this down into two components: the eastward component and the northward component.
The eastward component is given by:
eastward velocity = boat's speed * cos(30.0 degrees)
The northward component is given by:
northward velocity = boat's speed * sin(30.0 degrees)
Now, we can set up the equation for the boat's speed relative to the water:
relative speed = boat's speed - eastward velocity
Substituting the values we have:
relative speed = boat's speed - (boat's speed * cos(30.0 degrees))
Since we want the boat to move directly across the river, the northward component should be equal to the eastward velocity of the river, which is 4.0 m/s.
Therefore, the equation becomes:
4.0 m/s = boat's speed * sin(30.0 degrees)
Now, we can solve for the boat's speed by rearranging the equation:
boat's speed = 4.0 m/s / sin(30.0 degrees)
Using a calculator, we find:
boat's speed ≈ 8.0 m/s
So, the boat's speed relative to the water should be approximately 8.0 m/s.