The angle of projection for which maximum height is equal to the range covered will be? a)76 deg. b)tan-4. c)both

1 Answer
Mar 5, 2018

Arctan 4 = 76 degrees

Explanation:

The explanation involves using the position equation for both horizontal and vertical directions, evaluating height and range and setting them equal.
The position equation is
s(t) = s_"0" + v_"0" t + 1/2 a t^2
which for horizontal and vertical directions reduces to
s_"v"(t) = v_"0" sin x t - 1/2 g t^2
s_"h"(t) = v_"0" cosx t

solving for time of flight (tof) which is when s_"v"(t) = 0
0 = t (v_"0" sinx - g t / 2) ........ tof = 2 v_"0" sinx / g

Range = s_"h" ( tof) = v_"0" cosx ( 2 v_"0" sinx / g )
Range = s_"h" ( tof) = 2v_"0"^2 cosx (sinx / g )
Height = s_"v"( tof/2 ) = v_"0" sinx (v_"0" sinx / g ) - g/2 (v_"0" sinx / g )^2
Height = (v_"0" sinx)^2 / (2g)
Set these 2 expressions equal to get
(v_"0" sinx)^2 / (2g) = 2 v_"0"^2 cosx sinx /g
which simplifies to .....
sinx /2 = 2 cosx
tan x = 4
x = arctan 4 = 75.9 degrees