Runaway Plane

 

Launch worksheet

Launch solution 

The Scenario

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You are the crack security team leader at a covert base in a top secret location. You’ve almost finished your shift when you spy the prop plane accelerating down the runway without a pilot, out of control. You’ve got until the end of the runway to intercept the plane and smash your way into the cockpit to stop it before it takes off by itself!

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You’ve got two options. You can jump in the heavy lift helicopter and race to intercept the aircraft at the end of the runway. Or your second in command can use a free truck to try and make the intercept. You eyeball the angles as shown above in the diagram. Work out the travel distances required for the helicopter and the truck to make the intercept at the end of the runway.

Approach

For the helicopter intercept, you have a right-angled triangle with an angle of 80 degrees in one corner. To work out the travel distance for the helicopter, you need to work out which sides of the triangle are involved. 

In this case, it’s the side adjacent to the angle and also the hypotenuse along which the helicopter will be travelling. Using:

SOH CAH TOA

we can work out that we need to use cosine since it is “CAH” – adjacent and hypotenuse sides. 

\begin{aligned} cos(80)=\frac{adjacent}{hypotenuse}\\ cos(80)=\frac{100}{hypotenuse}\\ hypotenuse = \frac{100}{cos(80)}\\ hypotenuse = \frac{100}{cos(80)}\\ hypotenuse = 575.9 m \end{aligned}

 We can do the same for the truck – it’s the same type of triangle and angle relationship so we can also use cosine:

\begin{aligned} cos(70)=\frac{adjacent}{hypotenuse}\\ cos(70)=\frac{200}{hypotenuse}\\ hypotenuse = \frac{200}{cos(70)}\\ hypotenuse = \frac{200}{cos(70)}\\ hypotenuse = 584.8 m \end{aligned}

It turns out both the helicopter option and the truck option require almost exactly the same travel distance to make the intercept at the end of the runway.

Real Life Example – American Football (NFL)

American Football involves a lot of high skill level play. One of the most impressive plays occurs when a quarterback throws the ball to a touchdown player, which involves a process similar to the plane example above. The quarterback must take into account the expected trajectory of the runner, and throw the ball to intercept the player perfectly, without requiring the catching player to either slow down or to run faster than they possibly can.

Quarterbacks who do this perfectly time after time are paid incredible salaries in the tens of millions of dollars per year.

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