At-a-given-instant-two-cars-are-at-distances-600m-and-800m-from-the-point-of-intersection-of-the-straight-roads-crossing-at-right-angles-and-approaching-O-at-uniform-speeds-of-20-m-s-and-30-m-s-respe

Question Number 223270 by Tawa11 last updated on 20/Jul/25

At a given instant, two cars are at distances
600m and 800m from the point of intersection
of the straight roads crossing at right angles
and approaching O at uniform speeds of 20 m/s
and 30 m/s respectively.
Find the shortest distance between the cars
and the time taken to reach this position.

Answered by mr W last updated on 20/Jul/25

Commented by mr W last updated on 20/Jul/25

$${y}=\mathrm{800}+\frac{\mathrm{3}}{\mathrm{2}}\left({x}−\mathrm{600}\right)\:\Rightarrow\:\mathrm{3}{x}−\mathrm{2}{y}−\mathrm{200}=\mathrm{0} \\ $$$${d}_{{min}} =\frac{\mid\mathrm{3}×\mathrm{0}−\mathrm{2}×\mathrm{0}−\mathrm{200}\mid}{\:\sqrt{\mathrm{3}^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} }}=\frac{\mathrm{200}}{\:\sqrt{\mathrm{13}}}\approx\mathrm{55}.\mathrm{47}\:{m} \\ $$$${t}=\frac{\sqrt{\mathrm{600}^{\mathrm{2}} +\mathrm{800}^{\mathrm{2}} −\left(\frac{\mathrm{200}}{\:\sqrt{\mathrm{13}}}\right)^{\mathrm{2}} }}{\:\sqrt{\mathrm{30}^{\mathrm{2}} +\mathrm{20}^{\mathrm{2}} }}=\frac{\mathrm{360}}{\mathrm{13}}\approx\mathrm{27}.\mathrm{69}\:{s} \\ $$

Commented by Tawa11 last updated on 20/Jul/25

$$\mathrm{Thanks}\:\mathrm{sir}. \\ $$$$\mathrm{I}\:\mathrm{appreciate}. \\ $$

Answered by mr W last updated on 20/Jul/25

$${usual}\:{method}: \\ $$$$\Phi={d}^{\mathrm{2}} =\left(\mathrm{600}−\mathrm{20}{t}\right)^{\mathrm{2}} +\left(\mathrm{800}−\mathrm{30}{t}\right)^{\mathrm{2}} \\ $$$$\:\:\:\:=\mathrm{1300}{t}^{\mathrm{2}} −\mathrm{72000}{t}+\mathrm{1000000} \\ $$$$\frac{{d}\Phi}{{dt}}=\mathrm{2}×\mathrm{1300}{t}−\mathrm{72000}=\mathrm{0} \\ $$$$\Rightarrow{t}=\frac{\mathrm{72000}}{\mathrm{2}×\mathrm{1300}}=\frac{\mathrm{360}}{\mathrm{13}}\approx\mathrm{27}.\mathrm{69}\:{s} \\ $$$${d}_{{min}} =\sqrt{\left(\mathrm{600}−\mathrm{20}×\frac{\mathrm{360}}{\mathrm{13}}\right)^{\mathrm{2}} +\left(\mathrm{800}−\mathrm{30}×\frac{\mathrm{360}}{\mathrm{13}}\right)^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:=\frac{\mathrm{200}}{\:\sqrt{\mathrm{13}}}\approx\mathrm{55}.\mathrm{47}\:{m} \\ $$

Commented by Tawa11 last updated on 20/Jul/25

$$\mathrm{Thanks}\:\mathrm{sir}. \\ $$$$\mathrm{I}\:\mathrm{appreciate}. \\ $$

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