The variables in this question are:
1. Initial velocity of the arrow, v0, m/s
2. Angle with the horizontal, θ degrees
3. time to hit the target, t seconds.
There are two equations that govern these variables:
g is acceleration due to gravity = 9.8 m/s²
v0*cos(θ) = 450 m (horiz. distance) ....(1)
For the lower archer:
v0*sin(θ) + (1/2)(-g)t² = 150 m (vert. distance) .....(2)
For the upper archer:
v0*sin(θ) + (1/2)(-g)t² = -150 .....(3)
For the lower archer, the angle is determined, the remaining unknowns being v0 and t with two equations relating them, so there is a unique solution.
For the upper archer, the angle is undetermined. With three unknowns and two equations, there is an infinite sets of solutions. The archer will (arbitrarily) decide on the angle of attack, then the initial velocity can be determined.
In fact, the problem is very similar to the game called Gorilla included in the QBasic language with Windows 98.
I will proceed to solve the case of the lower archer.
From equation (1)
v0 = 450/(t*cos(θ))
substitute v0 in (2)
450/(t*cos(θ)) * sin(θ)*t + (1/2)(-g)t² = 150
Simplify to get:
Solve for t:
Substitute θ and t in (1) to get
v0 = 450/(t cos(θ))
Please check my derivation logic and calculations.
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