4. An airplane has an air velocity of 200 m/s [W]. The wind velocity relative to the ground is 60 m/s [N].
a. Determine the velocity of the airplane relative to the ground.
b. The airplane now faces a headwind of 60 m/s [E]. Calculate how long it takes the airplane to fly between two cities 300 km apart.
a. To determine the velocity of the airplane relative to the ground, we need to take into account the velocity of the airplane and the velocity of the wind.
Given:
Air velocity (Vair) = 200 m/s [W]
Wind velocity relative to the ground (Vwind) = 60 m/s [N]
To find the velocity of the airplane relative to the ground (Vplane):
1. Draw a vector diagram.
2. Start by placing the airplane velocity vector (200 m/s [W]).
3. Add the wind velocity vector (60 m/s [N]).
4. The airplane's velocity relative to the ground is the vector sum of the airplane velocity and the wind velocity (Vplane = Vair + Vwind).
Using vector addition, we can calculate the magnitude and direction of the resulting vector:
Magnitude:
Vplane = √((Vair)^2 + (Vwind)^2)
= √((200 m/s)^2 + (60 m/s)^2)
= √(40000 m^2/s^2 + 3600 m^2/s^2)
= √43600 m^2/s^2
≈ 208.7 m/s
Direction:
To determine the direction, we can use trigonometry. Let's consider the angle between the resultant vector and the west direction (W). We can use the tangent function to find this angle:
tan(theta) = (Vwind / Vair)
theta = arctan(Vwind / Vair)
= arctan(60 m/s / 200 m/s)
= arctan(0.3)
≈ 16.7°
Therefore, the velocity of the airplane relative to the ground is approximately 208.7 m/s [W 16.7° N].
b. To calculate the time it takes for the airplane to fly between two cities 300 km apart with a headwind of 60 m/s [E], we need to consider the airplane's velocity relative to the ground and the distance.
Given:
Distance = 300 km
Headwind velocity (Vwind) = 60 m/s [E]
First, we need to convert the distance to meters:
Distance = 300 km = 300,000 m
To calculate the time, we can use the formula:
Time = Distance / Velocity
Velocity of the airplane relative to the ground (Vplane) = Magnitude of Vplane = 208.7 m/s
Substituting the values:
Time = 300,000 m / 208.7 m/s
≈ 1437.5 s
Therefore, it takes approximately 1437.5 seconds or 23.96 minutes (rounded to the nearest minute) for the airplane to fly between the two cities 300 km apart with a headwind velocity of 60 m/s [E].