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
Pablocito
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

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