AIRSPEED A plane carries enough fuel for 20 hours of flight...

AIRSPEED A plane carries enough fuel for 20 hours of flight at an airspeed of 150 miles per hour. How far can it fly into a $30 \mathrm{mph}$ headwind and still have enough fuel to return to its starting point? (This distance is called the point of no return.)

Key Concepts

Relative Velocity

This concept involves computing an object’s effective speed when it is moving in a medium that also has its own motion. In aviation, the plane's airspeed relative to the air must be combined with the wind speed to obtain its ground speed, which is crucial for determining actual distances covered over time.

Ground Speed

Ground speed is the speed at which an aircraft moves relative to the ground. When flying with or against the wind, the effective speed increases or decreases accordingly. This adjustment is essential in planning flight times and distances, especially when considering headwinds or tailwinds.

Time-Distance-Speed Relationship

Fundamental to solving such problems is the relationship expressed by the equation distance = speed × time. By applying this formula to both legs of a round trip with differing speeds, one can set up equations that account for total available flight time to determine maximum reachable distances.

Fuel Constraints in Aviation

In scenarios where fuel capacity limits flight duration, fuel constraints become a critical factor. Managing fuel consumption involves calculating the time available for both the outbound and return segments, ensuring that the distance chosen does not exceed the limits imposed by the available fuel.

Transcript

00:02 So imagine a plane going from city a to city b and then back again, but we don't know the distance between the two cities.

00:12 We do know the rate of the plane's trip is affected by its air speed, which we can call it x.

00:25 And in this case, it's going to be 150 miles per hour.

00:29 And the wind speed will call y is 30 miles per hour.

00:37 We also know the total time is affected by this airspeed and wind speed.

00:46 To go from a to b is going to be different from b to a, depending on if it's with or against the wind.

00:53 But we can say, and we know that the time for, say, the first trip plus the time for the second trip is a total of 20 hours.

01:08 Plane has 20 hours worth of gas and we need to figure out the distance between these two cities that the plane can make.

01:16 So we have a couple of things to work through here, but we're going to use our distance equals rate times time and a system of equations to work through this.

01:25 Now essentially we would have our unknown distance and the rate going one direction is the air speed minus the wind speed.

01:38 It's going against the wind.

01:40 So air speed minus wind speed, 150 minus 30 times, let's just call this trip one or the time for trip one.

01:50 We don't know how long it takes because we don't know the distance.

01:54 And the same distance coming back.

01:57 Now it's getting to go with the wind.

02:00 So we add those two speeds together.

02:04 And this is however long it takes for trip two.

02:08 Now, we don't really want two separate variables here.

02:12 So this is where we can make a parameter and call these solving for the same variable...