The velocity of the plane relative to the ground, $\vec{v}_{p,g}$, is the vector sum of the plane's velocity relative to the air, $\vec{v}_{p,a}$, and the air's velocity relative to the ground (wind), $\vec{v}_{a,g}$.

$$ \vec{v}_{p,g} = \vec{v}_{p,a} + \vec{v}_{a,g} $$

First, we establish consistent units. Let the speed of the plane in still air be $v_{p,a} = 180$ km/h and the wind speed be $v_{a,g} = 10$ m/s. We convert the wind speed to km/h:

$$ v_{a,g} = 10 \frac{\text{m}}{\text{s}} \times \frac{3600 \text{ s}}{1 \text{ h}} \times \frac{1 \text{ km}}{1000 \text{ m}} = 36 \text{ km/h} $$

[Q1] Tailwind When flying with a tailwind, the wind vector $\vec{v}_{a,g}$ is in the same direction as the plane's velocity relative to the air $\vec{v}_{p,a}$. The magnitudes add directly.

$$ v_{p,g} = v_{p,a} + v_{a,g} $$

Substituting the values:

$$ v_{p,g} = 180 \text{ km/h} + 36 \text{ km/h} = 216 \text{ km/h} $$

[Q2] Headwind When flying with a headwind, the wind vector $\vec{v}_{a,g}$ is in the opposite direction to the plane's velocity relative to the air $\vec{v}_{p,a}$. The magnitudes subtract.

$$ v_{p,g} = v_{p,a} - v_{a,g} $$

Substituting the values:

$$ v_{p,g} = 180 \text{ km/h} - 36 \text{ km/h} = 144 \text{ km/h} $$

[Q3] Perpendicular Wind When flying perpendicular to the wind, the velocity vectors $\vec{v}_{p,a}$ and $\vec{v}_{a,g}$ are at a right angle to each other. The magnitude of the resultant velocity, the ground speed, is found using the Pythagorean theorem.

$$ v_{p,g} = \sqrt{v_{p,a}^2 + v_{a,g}^2} $$

Substituting the values:

$$ v_{p,g} = \sqrt{(180 \text{ km/h})^2 + (36 \text{ km/h})^2} $$ $$ v_{p,g} = \sqrt{32400 + 1296} \text{ km/h} = \sqrt{33696} \text{ km/h} $$

Simplifying the radical: $180 = 5 \times 36$.

$$ v_{p,g} = \sqrt{(5 \times 36)^2 + 36^2} = \sqrt{36^2 (5^2 + 1)} = 36\sqrt{26} \text{ km/h} $$