An airplane with an airspeed of 33.3 m/s is headed 30 degrees east of north in a wind blowing due west at 30 m/s. What is the ground speed of the plane?
Now I did the boat in the stream, same deal.
To find the ground speed of the plane, we need to combine the effect of the plane's airspeed and the wind's velocity.
First, let's break down the given information into components.
The airspeed of the plane, 33.3 m/s, is the magnitude of the velocity vector. Since the plane is headed 30 degrees east of north, we can split this velocity into two components:
- The northward component is given by: Vn = V * sin(θ)
where V is the airspeed and θ is the angle east of north.
Plugging in the values: Vn = 33.3 m/s * sin(30°)
Vn = 33.3 m/s * 0.5
Vn = 16.65 m/s (northward component)
- The eastward component is given by: Ve = V * cos(θ)
Plugging in the values: Ve = 33.3 m/s * cos(30°)
Ve = 33.3 m/s * 0.866
Ve = 28.7958 m/s (eastward component)
Next, let's consider the wind velocity. Since the wind is blowing due west, its speed would be the magnitude of the velocity vector. Thus, the westward component of the wind speed is 30 m/s.
Now, to calculate the ground speed, we need to combine the northward and eastward vectors of the plane's airspeed with the westward vector of the wind.
- The northward component of the ground speed would be: Vng = Vn - Vw
where Vw is the westward component of the wind speed.
Plugging in the values: Vng = 16.65 m/s - 30 m/s
Vng = -13.35 m/s (northward component of the ground speed)
- The eastward component of the ground speed would be: Veg = Ve
since the wind is not affecting the eastward component.
Plugging in the values: Veg = 28.7958 m/s (eastward component of the ground speed)
Finally, we can find the magnitude of the ground speed vector using the Pythagorean theorem:
Ground speed = √(Vng^2 + Veg^2)
Ground speed = √((-13.35 m/s)^2 + (28.7958 m/s)^2)
Ground speed = √(178.2225 m^2/s^2 + 829.52757 m^2/s^2)
Ground speed = √(1007.75 m^2/s^2)
Ground speed = 31.77 m/s (approximately)
Therefore, the ground speed of the plane is approximately 31.77 m/s.