A missile is fired from a launcher with an initial velocity of 250 m/s at a target located 500 m above the launcher and at a horizontal distance of 3000 m. Determine the following:

i) The possible firing angles required to hit the target.
ii) The maximum height reached for all possible angles.
iii) The time taken for the missile to hit the target for all possible firing angles.

(i) the range for angle θ is

R = v^2/g sin2θ

(ii) Naturally max height is reached at θ = 45°

(iii) given the range, and that horizontal velocity is constant, getting t is easy.

So, what do you get?

To answer these questions, we can use the equations of projectile motion. Let's solve each part step by step:

i) The possible firing angles required to hit the target:
We can use the equation for horizontal distance traveled by a projectile, which is given by:
Range = (initial velocity^2 * sin(2θ)) / g
where θ is the firing angle and g is the acceleration due to gravity (approximately 9.8 m/s^2).

In this case, the range is given as 3000 m. We can rearrange the equation to solve for the firing angle:
sin(2θ) = (Range * g) / (initial velocity^2)
θ = (1/2) * arcsin((Range * g) / (initial velocity^2))

Substituting the given values:
θ = (1/2) * arcsin((3000 * 9.8) / (250^2))
θ ≈ 21.32 degrees and 68.68 degrees

Therefore, the possible firing angles required to hit the target are approximately 21.32 degrees and 68.68 degrees.

ii) The maximum height reached for all possible angles:
To find the maximum height reached by the missile, we can use the equation for vertical displacement:
Vertical displacement = (initial velocity^2 * sin^2(θ)) / (2 * g)

For the firing angle of 21.32 degrees:
Vertical displacement = (250^2 * sin^2(21.32)) / (2 * 9.8)
Vertical displacement ≈ 294.39 meters

For the firing angle of 68.68 degrees:
Vertical displacement = (250^2 * sin^2(68.68)) / (2 * 9.8)
Vertical displacement ≈ 588.77 meters

Therefore, the maximum height reached for all possible firing angles is approximately 294.39 meters and 588.77 meters.

iii) The time taken for the missile to hit the target for all possible firing angles:
To find the time taken, we can use the equation for time of flight:
Time of flight = (2 * initial velocity * sin(θ)) / g

For the firing angle of 21.32 degrees:
Time of flight = (2 * 250 * sin(21.32)) / 9.8
Time of flight ≈ 15.62 seconds

For the firing angle of 68.68 degrees:
Time of flight = (2 * 250 * sin(68.68)) / 9.8
Time of flight ≈ 22.41 seconds

Therefore, the time taken for the missile to hit the target for all possible firing angles is approximately 15.62 seconds and 22.41 seconds.