# Difficult Velocity/Travel/Wind problem

• Aaron7
In summary, the conversation discusses an aeroplane's round trip between two airports A and B on a windless day and a day with wind blowing at an angle. The pilot maintains a constant speed relative to the air and always flies along the line AB and BA. The round trip time on the windless day is T0 and the round trip time on the windy day is T. Using the cosine rule and rearranging equations, the formula for T/T0 is determined to be 1/(kvcosθ + v√(1-k^2sinθ)), with a given answer of T/T0 = [(1-k^2(sin2θ))^0.5]/(1-k^2).
Aaron7

## Homework Statement

An aeroplane makes a round trip between two airports A and B located on the same line of longitude. On a windless day the aeroplane travels with constant speed v and the round trip time is T0. On the following day the same journey is made but there is a wind blowing at speed kv (k<1) at an angle z to the line AB. The pilot maintains the same speed v relative to the air and always flies along the line AB and BA. The round trip time is now T (neglecting take offs, landings and stopovers)

Find T/T0

See below

## The Attempt at a Solution

I drew a vector diagram for velocity with v_r going horizontally(new speed along AB when there is wind) and the components v (plane speed) and kv (wind speed).

I then used the cosine rule for v^2 = v_r^2 + (kv)^2 - (2 v_r kv cosθ)
Solved for v_r to get:
v_r = kvcosθ + v√(1-k2sinθ)

Now, T/T0 = v/v_r ,
so T/T0 = 1/(kvcosθ + v√(1-k2 sinθ))

-----------------------------------------------------------------------
The answer is given to be T/T0 = [(1-k2(sin2θ))0.5]/(1-k2)
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I have tried to rearrange to get the same answer but I think I have gone wrong somewhere.

Thanks for the help.

Hi Aaron7!
Aaron7 said:
An aeroplane makes a round trip between two airports A and B located on the same line of longitude.

(what does "r" stand for? )

There are two trips, AB and BA, with different speeds.

I got it Thanks for the help.

## 1. What is the difference between velocity and speed?

Velocity is a vector quantity that takes into account the direction of motion, while speed is a scalar quantity that only measures the magnitude of motion.

## 2. How do you calculate velocity in a difficult travel problem?

To calculate velocity, you need to divide the distance traveled by the time it took to travel that distance. The resulting value will have units of distance per time, such as meters per second or miles per hour.

## 3. How does wind affect velocity in a travel problem?

Wind can affect velocity by either adding to or subtracting from the overall speed and direction of an object. For example, a tailwind will increase the velocity of an airplane, while a headwind will decrease it.

## 4. What is the difference between average velocity and instantaneous velocity?

Average velocity is the overall displacement divided by the total time, while instantaneous velocity is the velocity at a specific moment in time. Average velocity can be calculated for a whole trip, while instantaneous velocity can vary throughout the trip.

## 5. How do you solve for velocity in a complex wind problem?

In a complex wind problem, it is important to break down the velocity into its components, including the horizontal and vertical components. From there, you can use trigonometry to calculate the overall velocity taking into account the direction and magnitude of the wind.

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