A boater wants to head straight across a river that has a current of 3km/hr. If the top speed of the boat in still water is 4km/hr, find the direction the boat must travel into the current to go straight across the river. How fast is the boat going straight across the river?
Assume current is moving due south(270o).
Vb = 4 - 3i.
tanA = -3/4 = -0.75
A = -36.9o = 36.9o South of East.
Vb = 4/cos36.9 = 5 km/h = Velocity of
boat.
The boat must travel 5 km/h @ 36.9o North of East.
To find the direction the boat must travel into the current to go straight across the river, we need to consider the effect of the current on the boat's motion.
Let's assume the current flows directly from left to right, and the desired direction of the boat is to go straight across the river from one bank to the other in a straight line.
To counteract the effect of the current and maintain a straight path, the boat must steer slightly upstream. This is because the current will push the boat downstream as it moves forward.
Now, let's find the speed at which the boat will be moving straight across the river, i.e., perpendicular to the current. We can use the Pythagorean theorem to calculate it.
The boat's speed relative to the water (in still water) is given as 4 km/hr. The current's speed is 3 km/hr. When the boat steers perpendicular to the current, the current's effect on the boat will be eliminated, and only the boat's speed will contribute to its motion across the river.
Using the Pythagorean theorem, we can find the speed of the boat straight across the river (let's call it B):
B² = (boat's speed in still water)² - (current speed)²
B² = 4² - 3²
B² = 16 - 9
B² = 7
Taking the square root of both sides, we find:
B = √7 km/h
So, the boat will be moving at a speed of √7 km/h straight across the river.
To find the direction the boat must travel into the current to go straight across the river, it needs to steer at an angle such that the vertical component of its velocity counteracts the downstream motion caused by the current. Since the boat's speed straight across the river is √7 km/h and the current's speed is 3 km/h, the boat's angle of travel should be the inverse tangent (arctan) of the ratio of the vertical component (3 km/h) to the horizontal component (√7 km/h), which will give us the angle in which the boat should steer.
Let's calculate that angle:
θ = arctan(vertical component / horizontal component)
θ = arctan(3 / √7)
Using a calculator, we find:
θ ≈ 55.4°
Therefore, the boat must travel at an angle of approximately 55.4 degrees into the current to go straight across the river, and it will have a speed of √7 km/h in that direction.