A boat moving at 5 mi/h is to cross a river in which the current is flowing at 3 mi/h. In what direction should the boat head to reach a point on the other bank of the river directly opposite the starting point?
X = Vc = 3 m/s.
Y = Vb = 5 m/s.
Tan A = Y/X = 5/3 = 1.66667.
A = 59o N. of E. = 31o E. of N.
Direction = 31o W. of N.
Well, if the boat heads directly towards the opposite bank, it's going to have a "current" situation! But don't worry, you don't want to be taking a "most drift" approach either. To reach the point directly opposite the starting point, the boat should head slightly upstream. It's all about finding the right balance between the boat's speed and the speed of the river current. So, tell that boat captain to set a course slightly upstream, and they'll reach their destination with no funny business!
To determine the direction the boat should head in, we need to consider the effect of the current on the boat's motion.
Let's assume that the river is flowing from left to right, and the boat wants to reach a point directly opposite its starting position.
Since the current is flowing at 3 mi/h, it will push the boat downstream as the boat tries to cross the river. To counteract the effect of the current, the boat needs to point its bow slightly upstream.
To find the angle the boat should head, we can use trigonometry. We'll create a right triangle, with the current's speed as the vertical leg (3 mi/h), the boat's speed as the hypotenuse (5 mi/h), and the angle between the hypotenuse and the horizontal leg as our unknown.
Using the trigonometric function tangent (tan), we can calculate the angle:
tan(angle) = opposite/adjacent
tan(angle) = 3/5
To find the angle, we can take the inverse tangent (arctan) of both sides:
angle = arctan(3/5)
Using a calculator, we find that the angle is approximately 30.96 degrees.
Therefore, the boat should head at an angle of approximately 30.96 degrees upstream, opposite to the direction of the current, to reach a point directly opposite its starting position on the other bank of the river.
To determine the direction the boat should head to reach a point directly opposite the starting point, we need to consider the effect of both the boat's speed and the river's current.
Here's how we can approach the problem step by step:
1. Draw a diagram: Start by drawing a diagram that represents the river, with the current indicated as an arrow. Label the starting point, the destination point on the other bank, and any other relevant points or distances.
2. Analyze the velocities: The boat has a velocity due to its own motion, which is 5 mi/h in this case. The river's current also has a velocity, which is 3 mi/h. To determine the effective velocity of the boat, we need to consider the vector sum of these velocities.
3. Apply vector addition: Since the boat's motion and the river's current are at right angles to each other (assuming the river flows straight across), we can use the Pythagorean theorem to find the boat's effective velocity.
a. The boat's velocity is 5 mi/h, which we can represent as a vector pointing straight across the river.
b. The river's current is 3 mi/h, which we can represent as a vector pointing downstream.
Using the Pythagorean theorem, we can find the magnitude of the boat's effective velocity:
effective velocity = sqrt(boat velocity^2 + current velocity^2) = sqrt(5^2 + 3^2) = sqrt(34)
4. Determine the direction: The direction the boat should head is given by the angle between the effective velocity vector and the direction straight across the river. We can find this angle using trigonometry.
a. The angle θ can be found using the tangent function: tan(θ) = opposite/adjacent = current velocity/boat velocity = 3/5.
b. Taking the inverse tangent of both sides, we find: θ = atan(3/5) ≈ 30.96 degrees.
5. Interpret the result: Based on the calculations, the boat should head at an angle of approximately 30.96 degrees upstream from straight across the river. This will allow it to counteract the effect of the current and reach a point directly opposite the starting point.
Remember to check your calculations and make sure the units of measurement are consistent throughout the process.