Projectile motion problem with only angle and distance given [closed]

Question

I have a cannon that shoots a projectile at an angle of 30°, the projectile hits a wall at 2.5m and it does so when the velocity only has a an horizontal component.

I need to find the initial speed, the time it takes for the projectile to hit the wall and at which height it hits the wall.

Since it says the projectile hits the wall only when there is an horizontal velocity, I suppose it does so when the vertical final velocity is 0.

I'm given the angle, the distance and I know $ g = 9.81m/s^2 $

$\begin{cases} x = x(0) +v(0)t \\y = y(0) + v(0)t -\frac{1}{2}gt^2 \end{cases} $

I need to find the initial velocity, but I'm missing time. I thought of using kinematics formulas and work on the x axis to find the time it takes the projectile to travel for 2.5m, but then I remembered that in projectile motion there is no acceleration on the x axis so I'm kind of stuck.

I was thinking of calculating the change in velocity on the y axis since I know its acceleration, so if I have $\Delta v = gt $ but I still don't have time.

Any hints on how I can find it? Thank you.

Answer 1

You are almost there. The two equations you provided are missing the $\theta$ term. It should be written:

$ x(t)=x_0+\left(v_0\cos\theta \right)t\\ y(t)=y_0+\left(v_0\sin\theta \right)t-\frac{1}{2}gt^2 $

You can also write the equations for the velocity, whose components are:

$ v_x(t)=v_0\cos\theta \\ v_y(t)=v_0\sin\theta - gt $

Now, "the projectile hits a wall [...] when the velocity only has a an horizontal component". Can you go from here?