A novice captain is pointing his ferryboat directly across the river at a speed of 15.7 mi/h. If he does not pay attention to the current that is headed downriver at 5.35 mi/h, what will be his resultant speed and direction? (Consider that the current and the boat’s initial heading are perpendicular to each other.)
Are you sure the captain is a novice? Despite the fact that he will land downstream from his starting point, his fastest path across the water is to head straight across.
Anyway I will pretend I am not a navigator and do what your book wants you to do.
For example:
He could be going East at 15.7 and South at 5.35
His resultant speed will be:
v = sqrt (15.7^2 + 5.35^2)
tan(angle ) = 5.35/15.7
where angle is the measure south of east
To determine the resultant speed and direction of the boat, we can use vector addition. The boat's speed and the current's speed are perpendicular to each other, so we can treat them as the sides of a right triangle.
Let's call the boat's speed "Vb" and the current's speed "Vc". According to the problem, Vb is 15.7 mi/h, and Vc is 5.35 mi/h.
To find the resultant speed, we can use the Pythagorean theorem:
Resultant Speed (Vr) = √(Vb² + Vc²)
Plugging in the values, we get:
Vr = √(15.7² + 5.35²) ≈ √(246.49 + 28.6225) ≈ √275.1125 ≈ 16.58 mi/h (rounded to two decimal places)
So, the resultant speed of the boat will be approximately 16.58 mi/h.
To determine the direction of the boat, we can use trigonometry. The current and the boat's initial heading are perpendicular, so the angle between them can be found using the inverse tangent function:
Theta = arctan(Vc / Vb)
Plugging in the values, we get:
Theta = arctan(5.35 / 15.7) ≈ arctan(0.340 (rounded to three decimal places)
Using a calculator, we find:
Theta ≈ 19.315 degrees (rounded to three decimal places)
So, the resultant direction of the boat will be approximately 19.315 degrees relative to its initial heading.