suppose you want to row your boat across a 3 mile wide river and row at a speed of 2 mi/hr. the current is flowing south at a rate of 1.2 mi/hr. at what angle upstream should you aim your boat so that it ends up going straight across?
Make a vector diagram, where the distance straight across is your resultant and represents the actual speed , the hypotenuse is 2 and the side opposite our angle is 1.2
sinØ = 1.2/2
Ø = 36.9°
To determine the angle at which you should aim your boat upstream, we can use the concept of vector addition.
Let's break down the velocity of your boat and the velocity of the river current into their horizontal and vertical components.
The velocity of your boat can be broken down as follows:
- Horizontal component: The distance you need to row across the river is 3 miles, and your rowing speed is 2 mi/hr, so the horizontal component of your velocity is 2 mi/hr.
- Vertical component: Since you want to counteract the southward river current, you need to row upstream. The current is flowing at a rate of 1.2 mi/hr, so the vertical component of your velocity is -1.2 mi/hr (negative because you are rowing against the current).
The velocity of the river current can be broken down as follows:
- Horizontal component: The river current is flowing horizontally, so its horizontal component is 0 mi/hr.
- Vertical component: The river current is flowing southward at a rate of 1.2 mi/hr, so the vertical component of the current's velocity is -1.2 mi/hr.
Now, let's add the vertical components of your boat's velocity and the current's velocity, which will give us the combined vertical velocity:
- Your boat's vertical velocity: -1.2 mi/hr (upstream)
- Current's vertical velocity: -1.2 mi/hr (downstream)
When you add these together, the vertical velocity cancels out: -1.2 mi/hr + (-1.2 mi/hr) = -2.4 mi/hr. Therefore, the combined vertical velocity is 0 mi/hr (canceled out).
Since we want your boat to go straight across the river, the combined horizontal velocity should be equal to the width of the river (3 miles). Therefore, we have:
- Your boat's horizontal velocity: 2 mi/hr
- Current's horizontal velocity: 0 mi/hr
When you add these together, we get the combined horizontal velocity: 2 mi/hr + 0 mi/hr = 2 mi/hr.
Now we can use trigonometry (specifically, tangent) to calculate the angle at which you should aim upstream. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side in a right triangle. In this case, the opposite side is the combined vertical velocity (0 mi/hr), and the adjacent side is the combined horizontal velocity (2 mi/hr).
Therefore, the tangent of the angle we're looking for is: tan(θ) = 0 mi/hr / 2 mi/hr = 0.
To find the angle itself, we can take the inverse tangent (arctan) of 0: θ = arctan(0). The arctan of 0 is 0.
Hence, the angle upstream that you should aim your boat is 0 degrees (or directly upstream).