Question

Long flights at midlatitudes in the Northern Hemisphere encounter the jet stream, an eastward airflow that can affect a plane’s speed relative to Earth’s surface. If a pilot maintains a certain speed relative to the air (the plane’s airspeed), the speed relative to the surface (the plane’s ground speed) is more when the flight is in the direction of the jet stream and less when the flight is opposite the jet stream. Suppose a round-trip flight is scheduled between two cities separated by 4000 km, with the outgoing flight in the direction of the jet stream and the return flight opposite it. The airline computer advises an airspeed of 1000 km/h, for which the difference in flight times for the outgoing and return flights is 70.0 min. What jet-stream speed is the computer using?

Answer 1

Answer:

The jet stream speed used by the computer is 142.$\overline {857142}$ km/h

Explanation:

The given parameters are;

The distance between the two cities the plane flies = 4,000 km

The difference in flight times for outgoing and return flights = 70.0 min

We note that 70 min = 70 min × 1 h/60 min = 7/6 h

The airspeed recommended by the airline computer = 1,000 km/h

Let 'a' represent the jet stream speed

The time it takes the plane moving in the same direction as the jet stream between the two cities, 't₁', is given as follows;

$t_1 = \dfrac{4000}{1000 + a}$

The time it takes the plane moving in the opposite direction as the jet stream between the two cities, 't₂', is given as follows;

$t_2 = \dfrac{4000}{1000 - a}$

The difference in flight times for outgoing and return flights, Δt = t₂ - t₁

Therefore, we have;

$\Delta t = t_2 - t_1 = \dfrac{4000}{1000 - a} - \dfrac{4000}{1000 + a} = \dfrac{7}{6}$

From which we get;

$-\dfrac{8000 \cdot a}{a^2 -1,000,000} = \dfrac{7}{6}$

By cross multiplying, we have;

-48,000·a = 7·a²- 7,000,000

∴ 7·a² + 48,000·a - 7,000,000 = 0

Factorizing with a graphic calculator gives;

(7·a - 1,000)·(a + 7,000) = 0

∴ a = 1,000/7, or a = -7000

Therefore, the jet stream speed the computer is using, a = 1,000/7 km/h = 142.$\overline {857142}$ km/h.