20. Miguel is driving his motorboat across a river. The speed of the boat in still water is 11 mi/h. The river flows directly south at 2 mi/h. If Miguel heads directly west, what are the boat’s resultant speed and direction?
To determine the boat's resultant speed and direction, we can use vector addition. The boat's resultant velocity (speed and direction) is the vector sum of its velocity in still water and the velocity of the river.
First, let's break down the velocities into their respective components. Since Miguel heads directly west, the boat's velocity in still water will have no north-south component, while the river's velocity will have no east-west component.
The boat's velocity in still water is 11 mi/h directly west. So, its components will be:
- East-West component: 11 mi/h
- North-South component: 0 mi/h
The river's velocity is 2 mi/h directly south. So, its components will be:
- East-West component: 0 mi/h
- North-South component: -2 mi/h
Now, we can add the components together to find the resultant velocity.
- East-West component: 11 mi/h + 0 mi/h = 11 mi/h
- North-South component: 0 mi/h + (-2 mi/h) = -2 mi/h
To find the magnitude of the resultant velocity (resultant speed), we use the Pythagorean theorem:
Resultant speed = √(east-west component^2 + north-south component^2)
Resultant speed = √(11 mi/h^2 + (-2 mi/h)^2)
Resultant speed = √(121 mi/h^2 + 4 mi/h^2)
Resultant speed = √125 mi/h^2
Resultant speed ≈ 11.18 mi/h
To find the direction of the resultant velocity, we use the inverse tangent (arctan) function:
Resultant direction = arctan(north-south component / east-west component)
Resultant direction = arctan(-2 mi/h / 11 mi/h)
Resultant direction ≈ -10.61° (measured counterclockwise from east)
Therefore, the boat's resultant speed is approximately 11.18 mi/h, and its direction is approximately 10.61° south of west.
make your sketch to get a right-angled triangle with sides 11 and 2.
The hypotenuse will be your resultant:
r^2 = 2^2 + 11^2 = 125
r = √125 = 5√3 or appr 11.18 mph
tanØ = 2/11
Ø = 10.3°
so it have a bearing of W 10.3° S