Minimal distance between two planes moving perpendicularly

[NikolasLund]

Homework Statement

Two planes fly at the same height with speeds v1 =
800 km/h and v2 = 600 km/h, respectively. The planes approach each other; at a certain moment of time, the plane trajectories are perpendicular to each other and both planes are at
the distance a = 20 km from the intersection points of their trajectories. Find the minimal distance between the planes during
their flight assuming their velocities will remain constant.

Relevant Equations

Hint: In the frame of the red plane, the blue plane moves along a line s which forms an angle α = arctan 3/4 with the horizontal dashed line in the figure. The distance of the red plane from this line is most conveniently found considering two similar right triangles, the larger of which is formed by the line s, and the two dashed lines in the figure.
The end result should be 4 km.

This problem is from Prof. Jaan Kalda's study guide to the IPHO. The problem can be solved by optimization, but there is apparently also a geometric approach, which is the one Kalda suggests. Initially I, being naive, tried to solve the problem by calculating the resulting distance the red plane needs to travel after t=1/40h (the time it takes the blue plane to reach the intersection). This, however, yielded a wrong result and prompted me to look at the hint, which didn't help me.

 

[Orodruin]

Did you try the suggested geometric approach? If not, why not? If yes, what was unclear about it?

The reason your approach gave the incorrect answer was that your presumption of the shortest distance occurring when one of the planes crossed the intersection point is incorrect.

 

[NikolasLund]

I attempted it, but could not figure out why the angle formed with the horizontal line is approx. 37 degrees. The resulting vector of the blue plane, in the frame of the red plane, I found to have a length of $a\sqrt 2$,being constructed by the already given vectors with lengths a. This obviously creates an isolesces triangle, where the angle is 45 degrees instead, which led me to the wrong answer of 3,5 km.

 

[Orodruin]

If two objects have velocities $\vec v_1$ and $\vec v_2$, respectively, what is the velocity of object 2 relative to object 1?

As for the optimisation route, did you try that one as well? If yes, what was the issues you had with it? Did you manage to write down the distance between the planes as a function of time?

 

[NikolasLund]

I managed the optimization with little problem, and got the correct answer there. As for the hint you gave regarding the geometric approach, I think I understand why the angle is $arctan 3/4$ ($|v_2|/|v_1|$ ?), but I am not sure how I should correctly use the relative velocity, or the given initial distance a.
PS Sorry for persisting with my lack of understanding, I am sure the problem is easier than I think.

 

[SammyS]

Homework Statement: Two planes fly at the same height with speeds v1 =
800 km/h and v2 = 600 km/h, respectively. The planes approach each other; at a certain moment of time, the plane trajectories are perpendicular to each other and both planes are at
the distance a = 20 km from the intersection points of their trajectories. Find the minimal distance between the planes during
their flight assuming their velocities will remain constant.
Relevant Equations: Hint: In the frame of the red plane, the blue plane moves along a line s which forms an angle α = arctan 3/4 with the horizontal dashed line in the figure. The distance of the red plane from this line is most conveniently found considering two similar right triangles, the larger of which is formed by the line s, and the two dashed lines in the figure.

Please provide the figure mentioned above.

 

[Lnewqban]

Two planes fly at the same height

That statement implies that both trajectories are contained in a common horizontal plane.
Also, the perpendicularity of the respective trajectories is comparable to the x and y axes of a Cartesian coordinate system.