A boat starts out at 20km/h in direction 30 degrees. However, the boat is blown off course by a 12km/h wind blowing from 80 degrees. What direction does the boat now travel in and how far does it travel in one hour?
I wish the math people would stay away from navigation. They are hopeless at distinguishing between bearing and heading and here they evidently think a boat moves with the wind like an airplane forgetting that the hull is in the water sort of like a car on a road in a crosswind slipping sideways a bit but not traveling sideways at the wind speed.
But anyway making their nutty assumption and assuming angles are clockwise from north:
speed east = 20 sin 30 - 12 sin 80
speed north = 20 cos 30 -12 cos 80
so
speed east = -1.82
speed north = 15.2
Tan Angle from north = -1.82 /15.2
so angle = -6.83
compass direction clockwise from north = 360 -6.83 = 353
speed = distance/hour
= sqrt(1.82^2+15.2^2) = 15.3 km/hour
To determine the direction the boat now travels in, we need to find the resultant velocity vector. We can do this by adding the vector representing the boat's velocity to the vector representing the wind's velocity.
To do this, we need to convert the velocities into their x and y components.
For the boat's velocity:
- Velocity (V) = 20 km/h at an angle of 30 degrees
We can find the x and y components of the boat's velocity using trigonometry:
- Vx (x-component) = V * cos(theta)
- Vy (y-component) = V * sin(theta)
Vx = 20 km/h * cos(30 degrees) = 20 km/h * (√3/2) = 10√3 km/h
Vy = 20 km/h * sin(30 degrees) = 20 km/h * (1/2) = 10 km/h
For the wind's velocity:
- Velocity (W) = 12 km/h at an angle of 80 degrees
Similarly, we can find the x and y components of the wind's velocity using trigonometry:
- Wx (x-component) = W * cos(theta)
- Wy (y-component) = W * sin(theta)
Wx = 12 km/h * cos(80 degrees) = 12 km/h * (0.1736) = 2.08 km/h
Wy = 12 km/h * sin(80 degrees) = 12 km/h * (0.9848) = 11.82 km/h
Now, to find the resultant velocity components, we add the respective x and y components:
- Rx (resultant x-component) = Vx + Wx
- Ry (resultant y-component) = Vy + Wy
Rx = 10√3 km/h + 2.08 km/h ≈ 15.86 km/h
Ry = 10 km/h + 11.82 km/h = 21.82 km/h
To find the magnitude and direction of the resultant velocity, we can use the Pythagorean theorem:
- R = sqrt(Rx^2 + Ry^2)
R = sqrt((15.86 km/h)^2 + (21.82 km/h)^2) ≈ 27.62 km/h
The direction of the resultant velocity vector can be found using the inverse tangent function:
- theta_r = arctan(Ry / Rx)
theta_r = arctan(21.82 km/h / 15.86 km/h) ≈ 53.44 degrees
Therefore, the boat now travels at a velocity of approximately 27.62 km/h in a direction of approximately 53.44 degrees.
To find the distance the boat travels in one hour, we can use the magnitude of the resultant velocity:
- Distance = R * time
Distance = 27.62 km/h * 1 hour = 27.62 km
Therefore, the boat travels approximately 27.62 km in one hour.
To determine the new direction and distance traveled by the boat, we need to consider the vector addition of the boat's velocity and the wind's velocity.
First, convert the given velocities to vector form.
The boat's initial velocity: magnitude = 20 km/h, direction = 30 degrees
Velocity of the wind: magnitude = 12 km/h, direction = 80 degrees
Next, draw a vector diagram representing the boat's initial velocity and the wind's velocity:
```
\
\
Velocity of \ \
the wind (12 km/h, \ \
80 degrees) <--------- Boat's initial velocity (20 km/h, 30 degrees)
/
/
/
```
Now, we can find the resultant vector by calculating the vector sum of the boat's initial velocity and the wind's velocity.
To do this, split the vectors into their vertical and horizontal components.
For the boat's initial velocity:
Vertical component = 20 km/h * sin(30 degrees)
Horizontal component = 20 km/h * cos(30 degrees)
For the wind's velocity:
Vertical component = 12 km/h * sin(80 degrees)
Horizontal component = 12 km/h * cos(80 degrees)
Now, add the horizontal and vertical components separately:
Vertical component of the resultant vector = vertical component of boat's initial velocity + vertical component of wind's velocity
Horizontal component of the resultant vector = horizontal component of boat's initial velocity + horizontal component of wind's velocity
Once you have the vertical and horizontal components of the resultant vector, you can find the magnitude and direction of the resultant vector using the Pythagorean theorem and trigonometry:
Resultant magnitude = sqrt(Vertical component of the resultant vector^2 + Horizontal component of the resultant vector ^2)
Resultant direction = arctan(Vertical component of the resultant vector / Horizontal component of the resultant vector)
Finally, with the resultant magnitude and direction, you can determine the direction the boat now travels in and the distance it travels in one hour.