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 and sketch for part (a)(i)

Let $\vec{v}_{h,a}$ be the velocity of the helicopter relative to the air. Since the helicopter is pointed west, and its magnitude is $250$ km/h, we can represent it as a vector of length proportional to $250$ km/h pointing to the left (west - direction).

Step2: Define vector symbols and sketch for part (a)(ii)

Let $\vec{v}_{a,g}$ be the velocity of the air relative to the ground. The wind is blowing from the east, so it is a vector of length proportional to $85$ km/h pointing to the right (east - direction).

Step3: Calculate velocity for part (b)

The velocity of the helicopter relative to the ground $\vec{v}_{h,g}$ is given by the vector addition $\vec{v}_{h,g}=\vec{v}_{h,a}+\vec{v}_{a,g}$. Since they are in opposite directions (west - east), the magnitude $v_{h,g}=250 - 85=165$ km/h west.

Step4: Calculate time for part (c)

We use the formula $t=\frac{d}{v}$, where $d = 1000.0$ km and $v = v_{h,g}=165$ km/h. So $t=\frac{1000.0}{165}\approx6.06$ h.

Answer:

(a)(i) $\vec{v}_{h,a}$, sketch: vector of length proportional to 250 km/h pointing west.
(a)(ii) $\vec{v}_{a,g}$, sketch: vector of length proportional to 85 km/h pointing east.
(b) 165 km/h west
(c) Approximately 6.06 h