Projectile Motion

You have been put in charge of designing a machine for the skeet
shooting championship. You have been told the following desired parameters:
- The clay targets must stay in the air for 9.17 seconds.
- The total horizontal distance travelled by the targets must be
550.46 meters.

Knowing that the machine will be put on the ground (0 meters high) and
the field is completely flat, you need to decide:
a) how much force must the targets be thrown into the air with, and
b) what angle needs to be used for the launch.
Consider .

There I added an attachment for more information about the problem. Any help would be great

a)  F = (m / t₀) 95.33

 b)  θ = 70.5º

Explanation:

This is a projectile launch, as they indicate the horizontal distance this is the range of the body,  let's use the expression for the range of the projectiles

            R = v₀² sin 2θ / g

            v₀² sin 2θ = R g

Where the range is  550.46 m

They also indicate the time that the air must remain, so this time is twice the time until reaching the maximum height.

        v_{y} = - g t

At the maximum height v_{y} = 0 and the initial speed on the axis and we can find it with trigonometry

         sin θ = v_{oy} / v_{o}

         v_{oy} = v_{o} sin θ

         v_{o} sin θ = g t

Let's write the two equations

             v_{oy}² sin 2θ = g R

             v_{o} sin θ = g t

 We solve our accusation system

              (G t / sin θ) 2 sin 2θ = g R

              g t² sin 2θ = R sin  θ

Let's use the trigonometric relationship

         sin 2θ = 2 sin θ cos θ

We substitute

           g t² (2 sin θ cos θ) = R sin θ

          Cos tea = R / 2 g t²

          θ = cos⁻¹ (R / 2g t²)

Let's calculate

          θ = cos⁻¹ (550.46 / (2  9.8  9.17² ))

          θ = 70.5º

a) Force can be  Newton's second law

On the x axis the speed is constant so the force on the axis is zero

In the y axis the acceleration we have is the acceleration of gravity, so the force that acts throughout the journey is the weight of the body.

To place the body in the air from the rest we can use the equation of the impulse

          F t = Δp = m v - m v₀

As kick from rest   v₀ = 0

Let's find the speed of the body

          = g t

          v_{o} = g t / sin 70.5

         v_{o} = 9.8 9.17 / sin 70.5

         v_{o} = 95.33 m / s

To encocorate the force we must suppose a firing time, which in general is very short, suppose that this time is to

           F = m v_{o} / t₀

           F = (m / t₀) 95.33

This is the outside that should be applied, as an example suppose a body of mass 1 kg⁵ ( m = 1 kg) and a trip time to = 0.1 s

           F = (1 / 0.1) 95.33

          F = 953.3 N

a displacement distance is size of displ