You are boating and heading directly west. The speed of the boat in still water is 36 mi/h. The river flows directly north at 15 mi/h. What is your resultant speed and directional?
result speed= sqrt(36^2+16^2)
direction(measured from the across direction) arc tan 36/16.
JAMES, CHECK YOUR 2-27-11,9:45AM POST.
To find the resultant speed and direction, we can use vector addition. Let's break down the problem into two components: the northward speed caused by the river flow and the westward speed of the boat itself.
1. Northward component:
The river flows directly north at a speed of 15 mi/h.
2. Westward component:
The boat's speed in still water is 36 mi/h when there is no external force, such as the river's current.
To find the resultant speed and direction, we need to find the vector sum of these two components.
Applying the Pythagorean theorem, the resultant speed can be calculated as follows:
Resultant speed = √(Northward component^2 + Westward component^2)
Calculating the northward component:
Northward component = River flow * cos(90°) [since the angle is 90° for a northward flow]
= 15 mi/h * cos(90°)
= 15 mi/h * 0 [cos(90°) = 0]
= 0 mi/h
Calculating the westward component:
Westward component = Boat speed * cos(0°) [since the angle is 0° for a westward heading]
= 36 mi/h * cos(0°)
= 36 mi/h * 1 [cos(0°) = 1]
= 36 mi/h
Calculating the resultant speed:
Resultant speed = √(0^2 + 36^2) [since the northward component is 0]
= √(0 + 1296)
= √1296
= 36 mi/h
Thus, the resultant speed is 36 mi/h.
To determine the direction, we can use trigonometry and find the angle between the resultant vector and the eastward direction.
Calculating the angle (θ):
tan(θ) = Northward component / Westward component
= 0 mi/h / 36 mi/h
= 0
Since tan(θ) = 0, this means the angle θ is 0° or 180°.
Since we are heading west, the direction is 180°. Therefore, the resultant speed is 36 mi/h westward.