How to chose the angle and initial speed to hit two objects

[Makonia]

1. The prIoblem statement, all variables and given/known data
You are the chief of the Angry Birds (Google it, if you don’t know the reference...it’s not important). You can shoot red birds from a catapult, and they will fly as projectiles under the effect of gravity. Gravity points downwards and has magnitude g = 9.80 m/s2 . Your goal is to hit some nasty green pigs, who have stolen your Angry Bird Eggs. You are able to adjust the angle θ and speed v of the projectile, as it leaves the catapult. The projectile leaves the catapult at a height h above the ground.
a) First, consider the case when a single green pig is placed on the ground, at a distance d from the foot of the catapult. Find a relation between the angle and speed required to hit it. Remember to draw a sketch of the situation.
b) Next consider the case where there are two green pigs. One on the ground at a distance d as before, the other at a distance l < d, and placed on a pole of height h. What should you choose for θ and v in order to hit both pigs 1 with the same red bird? Why must one require l < d? Remember to draw a sketch of the situation.

Homework Equations

On 1a) I came to the equation: Vi = sqrt((gd^2)/(2cos^(2)θ(h+d*tanθ)

The Attempt at a Solution

I think i have to somehow use the equation from a in order to find the V initial and angle but I don't know how.

Because the distance between the two objects isn't given and the projectile can hit with either a small angle and large innitial speed or a large angle and low speed. The fact that I need to find an expression for both of these when the distance isn't given is what's troubbeling me and any hints or pointers to which direction I should go next would be much appreaciated :) (sorry for my bad english, but I hope you understand)

This is the drawing that came with it btw
The task is given to the chapter about projectile motion and i think I'm supposed to use projectile motion to solve it.

 

[mfb]

Because the distance between the two objects isn't given

It is d-l.
You can adjust your equation for (1a) a bit to hit the pig on the pole. This will lead to a second equation involving v and θ, so you can solve the system for those two unknowns.

 

[Makonia]

It is d-l.
You can adjust your equation for (1a) a bit to hit the pig on the pole. This will lead to a second equation involving v and θ, so you can solve the system for those two unknowns.

So you sugest I replace the d in a with (d-l) and that's it? I mean it's okey that I get a new equation but how am I suposed to find both the angle and speed? Just by solving it for both of those one one side and that's it? How do I then know that it will also hit the pig on the ground after?

 

[mfb]

So you sugest I replace the d in a with (d-l) and that's it?

No, d-l is the distance between the two objects.
d is the distance between cannon and first object, you'll have to replace it by the distance between cannon and the other object.
Also take care of the height because now the object is on a pole (that will make the equation easier).

how am I suposed to find both the angle and speed?

You have two equations and two unknown parameters. That should be a situation you encountered before.

 

[Makonia]

No, d-l is the distance between the two objects.
d is the distance between cannon and first object, you'll have to replace it by the distance between cannon and the other object.
Also take care of the height because now the object is on a pole (that will make the equation easier).

You have two equations and two unknown parameters. That should be a situation you encountered before.

Makes sense so I've put the distance to l and h to 0 because the height is being shot from and hits the first target is the same. If I use the same way as with a and first find t from x=Vx*t and put this t in y = Vy*t -g/2*t^2 i get the equation : 0 = l*tanθ - g/2*(l^2)/(v^2*cos^2(θ)
The algebra is my problem i guess. If I want to find θ here it from these two eqations it would be difficult because i have both tanθ and cos^2θ. Would it help me if I solved the second eqotion for v and then set them equal to each other?

 

[mfb]

That's the approach I would try. The cosine terms should disappear then.