Question

a helicopter can fly at 250 km/h relative to the air, but the wind is blowing from the east at 85 km/h relative to the ground. the pilot points the helicopter west. assume the top of the page is north. (a) using proper vector symbols, record the following vectors and sketch the vectors. (i) the velocity of the helicopter relative to the air. (ii) the velocity of the air relative to the ground. (this is the wind) (b) using a vector diagram, calculate the velocity of the helicopter relative to the ground. (c) how much time would it take the helicopter to travel 1000.0 km with this tailwind?

Explanation:

Step1: Define vector symbols

Let $\vec{v}_{ha}$ be the velocity of the helicopter relative to the air and $\vec{v}_{ag}$ be the velocity of the air relative to the ground.

Step2: Record vectors for part (a)

(i)

The velocity of the helicopter relative to the air: $\vec{v}_{ha}= 250\ km/h$ west. In vector - component form, if we assume the positive x - axis is east and positive y - axis is north, $\vec{v}_{ha}=- 250\hat{i}\ km/h$.

(ii)

The velocity of the air relative to the ground: $\vec{v}_{ag}=85\ km/h$ east. In vector - component form, $\vec{v}_{ag}=85\hat{i}\ km/h$.

Step3: Calculate velocity of helicopter relative to ground for part (b)

The velocity of the helicopter relative to the ground $\vec{v}_{hg}$ is given by the vector addition $\vec{v}_{hg}=\vec{v}_{ha}+\vec{v}_{ag}$.
$\vec{v}_{hg}=-250\hat{i}+85\hat{i}=(-250 + 85)\hat{i}=-165\hat{i}\ km/h$. The magnitude of $\vec{v}_{hg}$ is $165\ km/h$ and the direction is west.

Step4: Calculate time for part (c)

We know that speed $v = 165\ km/h$ and distance $d = 1000.0\ km$. Using the formula $t=\frac{d}{v}$, we have $t=\frac{1000.0\ km}{165\ km/h}\approx6.06\ h$.

Answer:

(a)(i) $\vec{v}_{ha}=-250\hat{i}\ km/h$
(a)(ii) $\vec{v}_{ag}=85\hat{i}\ km/h$
(b) $165\ km/h$ west
(c) $t\approx6.06\ h$