A boat is going 30 mph with a direction of 100 degrees east of north. The water current is 15 mph with a direction of 22 mph. What is the magnitude of boat and direction
"The water current is 15 mph with a direction of 22 mph."
---> makes no sense to me
To find the magnitude (speed) and direction of the boat, we can use vector addition. The boat's velocity and the water current's velocity can be represented as vectors.
First, let's break down the velocities of the boat and the current into their horizontal (x) and vertical (y) components:
Boat velocity (Vb):
- Speed: 30 mph
- Direction: 100 degrees east of north
To find the horizontal and vertical components of the boat's velocity (Vbx, Vby), we can use trigonometry:
Vbx = Vb * sin(theta)
Vby = Vb * cos(theta)
Where theta is the angle between the boat's velocity and the positive x-axis (horizontal).
Vbx = 30 * sin(100)
Vby = 30 * cos(100)
Water current velocity (Vc):
- Speed: 15 mph
- Direction: 22 degrees
To find the horizontal and vertical components of the current's velocity (Vcx, Vcy), we can use trigonometry as well:
Vcx = Vc * sin(theta)
Vcy = Vc * cos(theta)
Where theta is the angle between the current's velocity and the positive x-axis (horizontal).
Vcx = 15 * sin(22)
Vcy = 15 * cos(22)
Next, we add the horizontal components and vertical components of the boat and the current vectors to get the resultant velocity (Vr):
Vrx = Vbx + Vcx
Vry = Vby + Vcy
Finally, we can calculate the magnitude and direction of the resultant velocity (Vr):
Magnitude (speed) of the boat and the current:
Magnitude (Vr) = sqrt(Vrx^2 + Vry^2)
Direction of the resultant velocity:
Direction = atan2(Vry, Vrx) * (180 / pi)
Now, let's calculate the values:
Vbx = 30 * sin(100) = -20.41 mph (negative because it's east of north)
Vby = 30 * cos(100) = 9.70 mph
Vcx = 15 * sin(22) = 5.53 mph
Vcy = 15 * cos(22) = 13.95 mph
Vrx = -20.41 + 5.53 = -14.88 mph
Vry = 9.70 + 13.95 = 23.65 mph
Magnitude (Vr) = sqrt((-14.88)^2 + (23.65)^2) = 28.08 mph (approximately)
Direction = atan2(23.65, -14.88) * (180 / pi) = 123.32 degrees (approximately)
Therefore, the magnitude of the boat's velocity and the current's velocity combined is approximately 28.08 mph, with a direction of 123.32 degrees east of north.