An airplane is flying at a velocity of 300km/h S30degrees W in a wind blowing at a velocity of 50km/h S20degrees E. What is the true speed and direction of plane?

converting your directions to standard trig notation

(x-axis is zero, counterclockwise)

S 30° W ----> 240°
S 20° E ----> 290°

300 km/h in S30W ---> vector(400cos240,400sin240)
50 km/h in S20E --> vector (50cos290 , 50sin 290)

resultant = (400cos240 + 50cos290 , 400sin240+50sin290)
= appr (-246.98, -393.339)

magnitude = √( (-246.98)^2 + (-393.339)^2)
= appr 464.5 km/h

direction:
tanØ = -393.339/-246.98) , Ø in III
Ø = appr 180 + 57.9 or 237.9°
or S 32.1° W

To determine the true speed and direction of the plane, we need to calculate the vector sum of the plane's velocity and the wind's velocity.

First, let's break down the given information:

- The plane's velocity: 300 km/h S30°W
- This means the plane is flying at a speed of 300 km/h in a direction located 30° west from the south direction.
- The wind's velocity: 50 km/h S20°E
- This means the wind is blowing at a speed of 50 km/h in a direction located 20° east from the south direction.

Now, to find the true speed and direction of the plane, we'll add the vectors using components. To do this, we'll convert the given velocities from polar form to Cartesian components. Let's start with the plane's velocity:

- The plane's velocity: 300 km/h S30°W
- To convert this to Cartesian components, we'll consider the x-axis as the east-west direction and the y-axis as the north-south direction.
- The x-component of the plane's velocity is given by: 300 km/h * sin(30°) = 150 km/h * 0.5 = 75 km/h
- The y-component of the plane's velocity is given by: 300 km/h * cos(30°) = 150 km/h * (√3/2) ≈ 150 km/h * 0.866 = 129.9 km/h
- Therefore, the plane's velocity can be represented as (75 km/h, 129.9 km/h).

Next, let's convert the wind's velocity into Cartesian components:

- The wind's velocity: 50 km/h S20°E
- To convert this to Cartesian components, we'll use the same direction convention as before.
- The x-component of the wind's velocity is given by: 50 km/h * cos(20°) = 50 km/h * (√3/2) ≈ 50 km/h * 0.939 = 46.95 km/h
- The y-component of the wind's velocity is given by: -50 km/h * sin(20°) = -50 km/h * 0.342 = -17.1 km/h (negative because it's blowing south)
- Therefore, the wind's velocity can be represented as (46.95 km/h, -17.1 km/h).

Now, we'll add the x-components and y-components of the plane's velocity and the wind's velocity to find the true speed and direction:

- The x-component of the true velocity is the sum of the x-components of the plane's velocity and the wind's velocity: 75 km/h + 46.95 km/h = 121.95 km/h
- The y-component of the true velocity is the sum of the y-components of the plane's velocity and the wind's velocity: 129.9 km/h - 17.1 km/h = 112.8 km/h

Finally, to find the true speed:
- The true speed is given by the magnitude of the true velocity vector, calculated using the Pythagorean theorem: √(121.95 km/h)^2 + (112.8 km/h)^2 ≈ √(14900.7025 km^2/h^2) ≈ 122.1 km/h

Therefore, the true speed of the plane is approximately 122.1 km/h.

To find the true direction:
- The true direction can be calculated using the inverse tangent function (tan⁻¹) of the y-component divided by the x-component of the true velocity vector: tan⁻¹(112.8 km/h / 121.95 km/h) ≈ tan⁻¹(0.925) ≈ 44.5°

Therefore, the true direction of the plane is approximately 44.5° relative to the east axis.

In summary, the true speed of the plane is approximately 122.1 km/h, and its direction is approximately 44.5° relative to the east axis.