Establish a relation between range and height

If a particle is thrown with a velocity u making an angle % with the horizontal, then range R is given by

R = (u2sin2%)/g _____(i)

also the maximum height attained by the particle is given by

h = (u2sin2%)/2g _____(ii)

thus, dividing eq. (ii) by eq. (i),

h/R = [(u2sin2%)/2g] / [(u2sin2%)/g]

= sin2%/sin2%

= sin2%/2sin%cos% [sin2% = 2sin%cos%]

= sin%/2cos%

= 1/2(tan%)

the ratio of the height to the range of a projectile is always given by half the tangent of the angle made by the projectile with the horizontal.

Nb:here % is taken insted of symbol theta