Superhero Struggle
A superhero protects the population of Zorg, a perfectly circular (2D) planet.
The superhero is currently standing at the top of the planet, upon receiving a distress call (at a place distinct from where the superhero is at). The superhero has his superpower which means that he can quickly reach the source of distress.
He can use the gravitational field to travel directly around the circumference of the planet, as shown in green below. Or, he can travel a certain distance directly towards the centre, travel in a circular arc around the centre of Zorg, and then travel the same certain distance directly towards the surface, as shown in grey below. However, he can't travel exactly to the centre as the nickel core would neutralise his powers. Also, he must travel clockwise around the planet to use the gravitational field.
Upon reaching the distress call, the superhero realised that the location of the distress call meant that it would have taken exactly the same amount of time to get there, no matter how he got there, even if he simply travelled around the circumference or went into the planet.
What is , the clockwise radian measure of the location of the distress call?
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1 solution
Let the outer radius be R and the inner radius be r. Then the distance travelled along the circumference is θ R , and the distance travelled by entering the planet first is 2 ( R − r ) + θ r . Equating the two gives:
θ R = 2 ( R − r ) + θ r
θ ( R − r ) = 2 ( R − r )
θ = 2