Kinematics Problem?

Question

Two straight railways converge to a level crossing at an angle α and two trains are moving towards the crossing with velocities u and v. If a and b are the intial distances of the trains from the crossing, show that their least distance apart will be:-
{(av-bu)sin α}/{(u^2+v^2-2uvcos α)^1/2}

Answer 1

This involves a bunch of messy algebra and bookkeeping.

Here's the technique:

Set T1=a-u*t from the crossing, and
T2=b-v*t from the crossing
The crossing is a vertex of a triangle with one adjacent side equal to T1 and the other adjacent side equal to T2
the third side is the distance of separation, D

With some trig:
D^2=T1^2*Sin^2 α + T2^2-2*T1*T2*cos α+T1*^2*cos^2 α
use sin^2=1-cos^2 to get

D^2=T1^2 + T2^2 - 2*T1*T2*cos α

substitute in T1 and T2, chug away at the algebra to get
D=sqrt(a^2-t*(2au+2bv-2cos α*(av+bu))+t^2*(u^2+v^2-2uvcos α)-2abcos α)
take the first derivative of dD/dt and set equal to zero and solve for t
t=(2au+2bv-2cos α*(av+bu))/(2(u^2+v^2-2uvcos α))

plug t into the equation for D and chug away to get your proof.

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