Projectile motion -- Maximum range it can travel inside a 2m tunnel

[rashida564]

Homework Statement

A particle is projected inside a tunnel which is 2 m high and must not hit the side or ceiling of the tunnel. If the initial speed is U show that the maximum range of the particle inside the tunnel
You may assume that the maximum range is attained when the projectile just grazes the roof of the tunnel.

Homework Equations

Δy=Usin(θ)t-½gt²
Δx=Ucos(θ)t

The Attempt at a Solution

Since the maximum range is attained when the project gazes the top of the roof then Δy=2m. U is know so the first equation I can use is 2=Usin(θ)-½gt² I assumed that vf=0 (even though I am not sure) 0=Usin(θ)t so I solve for t and substitute it into the first equation,then Idk what should I do I think it has something to do with Δx=Ucos(θ)t

 

[haruspex]

2=Usin(θ)-½gt²

You left something out, probably just a typo. But what exactly is t here?

0=Usin(θ)t

Eh? That would mean t=0.

something to do with Δx=Ucos(θ)t

Yes.
You have two equations with t in them, but you are not asked for a time. So what should you do as the next step?

 

[rashida564]

Sry for tge typos the dirst one I missed t and the second one it should be 0=Usin(theta)-gt

 

[rashida564]

I tried to eliminate t from one of them into the other x into the y but I didn't get anywhere. since theta is unknown

 

[haruspex]

I tried to eliminate t from one of them into the other x into the y but I didn't get anywhere. since theta is unknown

You now have a third equation (post #3) so you should be able to eliminate t and θ.