You and a friend are rowing a boat 8.5 KPH heading north. A cross current is moving 3.2 KPH at an angle of 38 degrees north of the west direction. How fast are you actually moving and what direction are you actually heading?
current north speed = 3.2 sin 38=1.97 N
current west speed = 3.2 cos 38 = 2.52 W
Total North speed = 8.5+1.97 = 10.47
Total west speed = 2.52
tan theta = 2.52/10.47
theta = 13.53 degrees west of north or
47.47 degrees north of west
Is the direction you are GOING (you are still HEADING north (on your compass)). Math teachers do not use proper navigation terminology. They confuse heading, bearing, and course made good.
v = 10.47/cos 13.53 =10.77 kph
Thank you!
To determine how fast the boat is actually moving and in what direction, we can use vector addition. The boat's speed and direction, as well as the cross current's speed and direction, must be taken into account.
First, let's break down the boat's speed and direction into its northward and westward components. The boat is moving at a speed of 8.5 KPH due north, so its northward component is 8.5 KPH. Since there is no eastward or westward component mentioned, we assume that the boat is not drifting sideways.
Next, let's break down the cross current's speed and direction into its northward and westward components. The cross current is moving at a speed of 3.2 KPH at an angle of 38 degrees north of the west direction. We can find the northward and westward components using trigonometry. The northward component is given by 3.2 KPH * cos(38°) and the westward component is given by 3.2 KPH * sin(38°).
Now, we can add the northward and westward components of both the boat and the cross current together to find the actual speed and direction of the boat's movement.
Boat's northward component: 8.5 KPH
Cross current's northward component: 3.2 KPH * cos(38°)
Boat's westward component: 0 KPH
Cross current's westward component: 3.2 KPH * sin(38°)
Actual northward speed: 8.5 KPH + 3.2 KPH * cos(38°)
Actual westward speed: 0 KPH + 3.2 KPH * sin(38°)
To find the magnitude (or speed) of the boat's actual movement, we can use the Pythagorean theorem:
Actual speed = √(Actual northward speed^2 + Actual westward speed^2)
To find the direction of the boat's actual movement, we can use the inverse tangent function:
Actual direction = atan(Actual westward speed / Actual northward speed)
By plugging in the given values and performing the calculations, we can determine the boat's actual speed and direction.