Show all steps (all the necessary algebra) in your work. Simply solving the prob
ID: 1653533 • Letter: S
Question
Show all steps (all the necessary algebra) in your work. Simply solving the problem with Wolfram or similar software will not give you any credit for these problems. Suppose you launch an object from the ground at angle delta and it lands back on the ground a distance d beyond a target. You fire another projectile from the same location with the same initial velocity, but the launch angle is gamma. This time the object lands a distance c short of the target Prove that the correct angle theta¸ required to hit the target can be expressed as theta = 1/2 arcs in c sin delta + d sin gamma/c + d. If you use an equation from the textbook that was not derived in class, you must include its derivation in your solution for full credit.Explanation / Answer
Let x be the target distance
then
usinmg horizontal range in case of two dimesional projectile is
X = [Vo^2*sin(2*theta)] / g .............(1)
for an angle delta
(X+d) = [Vo^2*sin(2*delta)] / g ...........(2)
for an angle gamma
(X-c) = [Vo^2*sin(2*gamma)] / g ...........(3)
substituing (1) in (2)
[Vo^2*sin(2*theta)] / g + d = [Vo^2*sin(2*delta)] / g
[Vo^2*sin(2*theta)] + gd = [Vo^2*sin(2*delta)]
Vo^2*(sin(2*delta) - sin(2*theta)) = gd
Vo^2 = gd/(sin(2*delta) - sin(2*theta))...(4)
sunstituting (1) in (3)
[Vo^2*sin(2*theta)] / g - c = [Vo^2*sin(2*gamma)] / g
[Vo^2*sin(2*theta)] - gc = [Vo^2*sin(2*gamma)]
Vo^2*(sin(2*gamma) - sin(2*theta)) = -gc
Vo^2 = -gc/(sin(2*gamma) - sin(2*theta)).....(5)
from (4) and (5)
gd/(sin(2*delta) - sin(2*theta)) = -gc/(sin(2*gamma) - sin(2*theta))
d/(sin(2*delta) - sin(2*theta)) = -c/(sin(2*gamma) - sin(2*theta))
d*(sin(2*gamma) - sin(2*theta)) = -c*(sin(2*delta) - sin(2*theta))
c*sin(2*delta) + d*sin(2*delta) = (c+d)*sin(2*theta)
sin(2*theta) = [c*sin(2*delta) + d*sin(2*delta)] /(c+d)
if delta is too small or large sin(2*delta) = sin(2*delta)
similarly if gamma is too small or larger then sin(2*gamma) = sin(gamma)
then
sin(2*theta) = [c*sin(delta) + d*sin(delta)] /(c+d)
2*theta = arcsin {[c*sin(delta) + d*sin(delta)] /(c+d)}
theta = (1/2) arcsin {[c*sin(delta) + d*sin(delta)] /(c+d)}