a small blimp has an air speed of 24km/h on a heading of 42°. The wind's speed is 9 km/h and its direction is 312°. Find the ground speed and drift angle of the blimp.
If R is the resultant vector, then
R = (24cos42,24sin42) + (9cos312,9sin312)
= (17.8355, 16.059) + (6.02218, -6.6883)
= (23.85766, 9.3707)
magnitude = √(23.85766^2 + (9.3707)^2) = appr 25.63 km/h
direction: Ø = tan^-1(9.3707/23.85766) = appr 21.44°
or
make a sketch and see that the two vectors are at right angles
to each other, so
R^2 = 24^2 + 9^2
R = appr 25.63, same as before
larger angle of right-angled triangle:
tan A = 24/9
A = 69.44°
direction angle = 69.44 + 312 = 381.44° which is coterminal with 21.44°
the same as before.
So if the heading was 42° and the actual direction is 21.44, what is the drift ?
All angles are measured CW from +Y-axis.
Vb + 9[312o] = 24km[42o].
Vb + 9*sin312+i9*Cos312 = 24*sin42 + i24*Cos42,
Vb - 6.69 + 6.02i = 16.1 + 17.8i,
Vb = 22.79 + 11.78i = 25.7km[62.7o]. = Velocity and direction of the blimp(gnd. speed).
Tan A = X/Y = 22.79/11.78.
Drift = 62.7 - 42. = 20.7o.
To find the ground speed and drift angle of the blimp, we can use vector addition.
1. Resolve the air speed and wind speed vectors into their horizontal (x) and vertical (y) components:
Air speed:
- Air speed magnitude = 24 km/h
- Air speed angle = 42°
- Air speed x-component = air speed magnitude * cos(air speed angle)
- Air speed y-component = air speed magnitude * sin(air speed angle)
Wind speed:
- Wind speed magnitude = 9 km/h
- Wind speed angle = 312°
- Wind speed x-component = wind speed magnitude * cos(wind speed angle)
- Wind speed y-component = wind speed magnitude * sin(wind speed angle)
2. Add the x-components and y-components of the air speed and wind speed vectors:
Ground speed x-component = air speed x-component + wind speed x-component
Ground speed y-component = air speed y-component + wind speed y-component
3. Calculate the magnitude of the ground speed vector using Pythagorean theorem:
Ground speed magnitude = √(ground speed x-component^2 + ground speed y-component^2)
4. Calculate the drift angle using the inverse tangent function:
Drift angle = arctan(ground speed y-component / ground speed x-component)
Now let's calculate the values:
Air speed x-component = 24 km/h * cos(42°) = 16.532 km/h
Air speed y-component = 24 km/h * sin(42°) = 15.544 km/h
Wind speed x-component = 9 km/h * cos(312°) = -5.644 km/h
Wind speed y-component = 9 km/h * sin(312°) = -4.632 km/h
Ground speed x-component = 16.532 km/h + (-5.644 km/h) = 10.888 km/h
Ground speed y-component = 15.544 km/h + (-4.632 km/h) = 10.912 km/h
Ground speed magnitude = √(10.888 km/h)^2 + (10.912 km/h)^2 = 15.455 km/h
Drift angle = arctan(10.912 km/h / 10.888 km/h) = 45.04°
Therefore, the ground speed of the blimp is approximately 15.455 km/h and the drift angle is approximately 45.04°.
To find the ground speed and drift angle of the blimp, we can first break down the given information into the blimp's velocity components.
The air speed of the blimp is given as 24 km/h, and its heading is 42 degrees. This tells us that the blimp is moving at 24 km/h in the direction of 42 degrees from the reference direction (usually North).
The wind's speed is given as 9 km/h, and its direction is 312 degrees. This means that the wind is blowing at 9 km/h in the direction of 312 degrees from the reference direction.
To find the ground speed, we need to add the velocity components of the blimp and the wind.
Step 1: Convert the blimp's heading to the reference direction (North) by subtracting it from 90 degrees. In this case, 90 - 42 = 48 degrees. This angle represents the direction of the blimp relative to North.
Step 2: Convert the wind's direction to the reference direction (North) using the same method. 90 - 312 = 222 degrees. This angle represents the direction of the wind relative to North.
Step 3: Calculate the horizontal (x) and vertical (y) components of the blimp's velocity and the wind's velocity using trigonometry:
Blimp's x-component: blimp_speed * cos(blimp_heading)
Blimp's y-component: blimp_speed * sin(blimp_heading)
Wind's x-component: wind_speed * cos(wind_direction)
Wind's y-component: wind_speed * sin(wind_direction)
Step 4: Calculate the ground speed and drift angle by adding the horizontal components and vertical components:
Ground speed = sqrt((blimp_x + wind_x)^2 + (blimp_y + wind_y)^2)
Drift angle = arctan((blimp_y + wind_y) / (blimp_x + wind_x))
Now we can substitute the given values into the equations to find the answers:
Blimp's x-component: 24 km/h * cos(48 degrees) = 15.63 km/h
Blimp's y-component: 24 km/h * sin(48 degrees) = 18.36 km/h
Wind's x-component: 9 km/h * cos(222 degrees) = -6.83 km/h (negative because it is blowing in the opposite direction)
Wind's y-component: 9 km/h * sin(222 degrees) = -3.53 km/h (negative because it is blowing in the opposite direction)
Ground speed = sqrt((15.63 km/h - 6.83 km/h)^2 + (18.36 km/h - 3.53 km/h)^2) = sqrt(81.8 + 242.1) = sqrt(323.9) = 18.0 km/h (rounded to 1 decimal place)
Drift angle = arctan((18.36 km/h - 3.53 km/h) / (15.63 km/h - 6.83 km/h)) = arctan(14.83 km/h / 8.80 km/h) = arctan(1.68) = 58.4 degrees (rounded to 1 decimal place)
Therefore, the ground speed of the blimp is approximately 18.0 km/h, and the drift angle is approximately 58.4 degrees.