A straight river flows at a speed of 20km/h. A boater starts at the south shore and heads in ths direction of 70degrees from the shore. The boat has a speed of 20km/h in still water. Find the true speed and direction of the boat.
Vb = 20km @ 70o + 20km @ 0o.
Vb = X + Yi.
Vb = (20*cos70*20) + (20*sin70)i
Vb = 26.84 + 18.79i
tanA = Y/X = 18.79/26.84 = 0.70022
A = 35o = Direction.
Vb = X/cosA = 26.84/cos35 = 32.8 km/h =
Velocity of the boat.
To find the true speed and direction of the boat, we need to consider both the speed and direction of the river's flow and the speed of the boat in still water.
Let's break down the problem into components:
1. River's flow: The river flows at a speed of 20 km/h.
2. Boat's speed in still water: The boat has a speed of 20 km/h.
3. Boat's heading: The boat is heading at an angle of 70 degrees from the shore.
To find the true speed and direction of the boat, we can use vector addition.
Step 1: Break down the river's flow and the boat's heading into their respective horizontal and vertical components.
For the river's flow:
- Horizontal component: The river's flow does not have any horizontal velocity since the flow is perpendicular to the boat's heading.
- Vertical component: The vertical component of the river's flow would be 20 km/h since the river is flowing straight.
For the boat's heading:
- Horizontal component: To find the horizontal component of the boat's heading, we need to multiply the boat's speed in still water by the cosine of the angle. In this case, it would be 20 km/h * cos(70°).
- Vertical component: To find the vertical component of the boat's heading, we need to multiply the boat's speed in still water by the sine of the angle. In this case, it would be 20 km/h * sin(70°).
Step 2: Add the respective horizontal and vertical components of the river's flow and the boat's heading to get the total:
- Horizontal component: Since both the river's flow and the boat's heading do not have horizontal components, the total horizontal component would be 0.
- Vertical component: Add the vertical components of the river's flow and the boat's heading. It would be 20 km/h + (20 km/h * sin(70°)).
Step 3: Using the horizontal and vertical components of the total, calculate the true speed and direction of the boat.
- True speed: The true speed of the boat can be found using the Pythagorean theorem: square root of (horizontal component squared + vertical component squared).
- Direction: The direction of the boat can be found using the inverse tangent function: atan(vertical component / horizontal component).
Plug in the values and calculate to find the true speed and direction of the boat.