A particle is thrown over a triangle from one end of horizontal and grazing the vertex falls at other end - Physics - Motion In A Plane

Question

A particle is thrown over a triangle from one end of horizontal and
grazing the vertex falls at other end of base If A and B be the
base angles and C the angle of projection prove tan C= tan A + tan
B

Jyoti Pant · Accepted Answer

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Taking horizontal base ab as x-axis and a as origin
Drawing cd perpendicular to ab
 Let cd=had=hcot Adb=hcot BThus, ab=h (cot A+cot B)--(i)Coordinates of c are (ac, cd)i
e
 (hcot A,h)Equation of the path is,y=x tan θ-gx22u2cos2θThus, c (hcotA,h) lies on it
So,h=h cot A tan C-gh2cot2A2u2cos2Ci e
 1=cot A
tan C-ghcot2A2u2cos2C---(ii)Horizontal range=ab=u2sin 2CgTherefore, h(cotA+cot B)=u2sin 2Cg   [using (i)]u2=gh(cotA+cot B)2sin 2CFrom (ii),1=cot AtanC-ghcot2A2cos2C
sin 2Cgh(cotA+cot B)=cot AtanC-cot2A
 2sinCcosC2cos2C(cotA+cot B)=cot AtanC-cot2A
tan Ccot A+cot B=tan C[ cot A-cot2A(cotA+cot B)]=tan C
(cotAcot B)(cotA+cot B)Thus, tan C=(cotA+cot B)(cotAcot B)=1cot A+1cot BOr tan C=tan A+tan B

A particle is thrown over a triangle from one end of horizontal and grazing the vertex falls at other end of base . If A and B be the base angles and C the angle of projection prove tan C= tan A + tan B