How to Calculate the Correct Angle and Time to Cross a River in a Powerboat?

[chaudown]

A 310-m-wide river has a uniform flow speed of 1.6 m/s through a jungle and toward the east. An explorer wishes to leave a small clearing on the south bank and cross the river in a powerboat that moves at a constant speed of 4.4 m/s with respect to the water. There is a clearing on the north bank 11 m upstream from a point directly opposite the clearing on the south bank. (a) At what angle, measured relative to the direction of flow of the river, must the boat be pointed in order to travel in a straight line and land in the clearing on the north bank? (b) How long will the boat take to cross the river and land in the clearing?


I don't know whether the 4.4 m/s is a magnitude or the y-component.

 

[Hootenanny]

An explorer wishes to leave a small clearing on the south bank and cross the river in a powerboat that moves at a constant speed of 4.4 m/s with respect to the water.

I don't know whether the 4.4 m/s is a magnitude or the y-component.

What is speed in relation to velocity?

 

[Moetasim]

I will assume it is the magnitude since it is the only speed given in the problem.

(a) To determine the angle at which the boat must be pointed, we can use the law of sines. Let x be the angle relative to the direction of flow of the river. Then, we have the following triangle:

sin x / 1.6 = sin (90-x) / 4.4

Solving for x, we get x = 48.5 degrees. This means that the boat must be pointed at an angle of 48.5 degrees relative to the direction of flow of the river in order to travel in a straight line and land in the clearing on the north bank.

(b) To calculate the time it will take for the boat to cross the river and land in the clearing, we can use the formula d = vt, where d is the distance, v is the velocity, and t is the time. The distance the boat must travel is the hypotenuse of the triangle formed by the boat's path. Using the Pythagorean theorem, we get:

d = √(11^2 + 310^2) = 310.5 m

Plugging in the values, we get:

310.5 = 4.4t

Solving for t, we get t = 70.5 seconds. Therefore, it will take the boat approximately 70.5 seconds to cross the river and land in the clearing on the north bank.