Boat Velocity in a River: Angle for 4.0km/h & 1.8km/h

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In summary, the boat must be pointed at a 27 degree angle upstream to travel straight across the river, taking into account the current of 1.8 km/h. This can be found by using the equation sin-1(1.8/4) and the trigonometric function SOH CAH TOA. This is because the boat's velocity must equal the vector sum of its velocity in relation to the water in order to travel straight across the river.
  • #1
mikefitz
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A boat that can travel at 4.0 km/h in still water crosses a river with a current of 1.8 km/h. At what angle must the boat be pointed upstream to travel straight across the river? In other words, in what direction is the velocity of the boat relative to the water?

I have already found the answer due to the equation: sin-1(1.8/4)

The answer is 27 degrees upstream.

Reading through my book, I understand this has something to do with the boat's velocity equaling the vector sum of it's velocity in relation to the water?

Something like that. Can any of you clarify the relationship between the boat and water speeds, and why this equation yielded a correct answer? Thanks!
 
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  • #2
the boat is trying to go straight across, but the water is pushing it to the side at 1.8km/h. so to compensate for the current, the boat has to travel at 27 degrees upstream. (picture a triangle).

~Amy
 
  • #3
I understand that, but how do figure out that sin-1(1.8/4) is the method of reaching the 27 degree conclusion?
 
  • #4
if you draw a right angled triangle, you ll see that the hypotenuse is 4 km/h and the opposite side is 1.8 km/h.

so using the trig thing SOH CAH TOA (let me know if you're not familiar with this), you'll see that you only have the hypo. side and the opposite side so the only one you can use is the sine function (SOH). you want to find the opposite angle.

sin = opposite/hypo. = 1.8/4 = 0.45 sin-1 = 27 degrees.

~Amy
 

1. How is boat velocity calculated in a river?

The boat velocity in a river is determined by the vector sum of two velocities: the velocity of the boat relative to the water and the velocity of the water relative to the shore. This is known as the relative velocity formula: Vb = Vw + Vr, where Vb is the boat velocity, Vw is the velocity of the water, and Vr is the velocity of the boat relative to the water.

2. What is the significance of the angle in calculating boat velocity in a river?

The angle represents the direction in which the boat is traveling relative to the direction of the water flow. This angle is crucial in determining the magnitude and direction of the boat's velocity in the river.

3. How does the angle affect the boat's velocity in a river?

The angle of the boat's direction relative to the water flow determines the magnitude and direction of the boat's velocity in the river. A larger angle will result in a slower boat velocity, while a smaller angle will result in a faster boat velocity.

4. What is the practical application of understanding boat velocity in a river?

Understanding boat velocity in a river is important for various activities such as navigation, recreational activities like rafting and canoeing, and for predicting travel times. It can also be useful in engineering and hydrology for designing structures and predicting water flow.

5. How can the angle and boat velocity in a river be measured?

The angle and boat velocity in a river can be measured using instruments such as a compass and a speedometer. Alternatively, it can also be calculated using the relative velocity formula and other variables such as the boat's speed relative to the water and the water's speed relative to the shore.

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