Imagine that you are the pilot of the light aircraft, which is capable of cruising at a steady speed of 300 kph in still air. There is enough fuel on board to last for four hours. Consider firstly the situation on a perfectly calm day, i.e. no wind. A graphical approach will be used to solve the problem. A graph will be used to show how distance from the airfield varies with time, i.e. plot Distance versus Time.

Mark a point on the graph to represent the plane's initial position. Draw a line to represent the outward speed of 300 kph. What feature of the graph gives the speed directly? Assuming that the plane arrives back at the airfield exactly four hours after it left, mark a point on the graph to represent its final position. Draw a line to represent the return speed of 300 kph. Find the maximum distance that the pilot can fly from the airfield, and still be sure that there is enough fuel to make a safe return journey - this is "the point of no return", and state what time it takes to reach this point.

Mark this point of no return clearly on your graph. Construct two linear equations that when solved simultaneously would give you the 'point of no return'. Check your results with the results above.

Task 2:

Consider now the situation when it is no longer calm, i.e. there is a wind blowing.

Consider now the case when the plane takes off from the airfield and, on the outward journey, is helped along by a 50 kph wind (i.e. a "tail wind") which increases cruising speed relative to the ground to 350 kph. Suddenly the pilot realises that on the return journey the plane will be flying into the wind and will therefore slow down to 250 kph. Again, there is only enough fuel on board to last for four hours.

Use the graph to show an outward speed of 350 kph.

Use the graph to show a return speed of 250 kph.

Mark on the graph the new "point of no return" and state its distance from the airfield.

How long will it take to reach this point?

Consider another wind speed, this time a 50 kph head wind on the outward journey. Repeat the calculations above, and mark the new "point of no return" on the graph.

Repeat these 2 calculations with at least 2 other wind speeds of your own choice, showing calculations and plotting these points on the graph also.

If these points of no return were joined to form a smooth curve, what shape do you think the curve would have?

Task 3:

Consider the situation where:

The wind speed is w kph, The "point of no return", d km, is the distance from the airfield, and The time at which the pilot should turn around is t hours.

You can assume that there is only enough fuel on board to last for four hours, and the plane is capable of cruising at a steady speed of 300kph in still air.

You need to express d in terms of t only. You can use the steps below, or you can use an alternate method if you prefer. Write down two expressions for the outward speed of the aircraft, one involving w and one involving d and t. Write down two expressions for the homeward speed of the aircraft, one involving w and one involving d and t. Express d in terms of only t by eliminating w from the two resulting equations. Does this explain the pattern made by your "points of no return"? Why/why not?

Task 4:

What wind speeds - head or tail - would prevent the plane from completin journey? Consider the plane being capable of cruising at a steady speed of a kph. Rep part 3. Express d in terms of t and a.

State any assumptions which need to be applied.