Prove the postulate below?

Question

The drivers of a train moving at a speed #v_1# sight another train at a distance #d# ahead of them on the same track, moving with slower speed #v_2#. 1st driver pulles the brake to create a retardation #r#. Show that if #d > (v_1 - v_2)^2/(2r)#, there will be no collision.

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Answer 1 · 2017-05-07T15:52:09

Using the equation of motion: #s = v_it + 1/2 a t^2#

Using the position of the first train at #t = 0# as the Origin, the respective displacement of the trains at time #t# is:

#p_1(t) = v_1 t - 1/2 r t^2#

#p_2(t) = d + v_2 t #

#p_1(t) = p_2(t) implies v_1 t - 1/2 r t^2 = d + v_2 t #

This is a quadratic:

# r/2 t^2 + (v_2 - v_1) t + d = 0#

In the quadratic equation, we force the discriminant (ie #b^2 - 4ac#) to be negative to ensure there are no real solutions,

#b^2 < 4ac implies (v_2 - v_1)^2 < 4 r/2 d#

# implies d > (v_2 - v_1)^2/(2r)#