Help Solving Calculus/Mechanics Question on Particle in Gravity Field

[zebrastripes]

I'm really struggling to get my head around some of the mechanics applications in my calculus module and would really appreciate it if somebody could help me get my head around it.

I have this question:

A particle P of constant mass m has a position vector:

r=x(t)i+y(t)j,

and moves in a uniform gravitational field -gj. At time t=0, P is at the origin of the coordinate system and is projected with speed U at an angle 0≤θ≤∏ to the vector i.

We are given that r''=-gj.
Hence show that y=0 at a time t=τ>0 where τ=2Usinθ/g.

So, I have that y''(t)=-g,
so y'(t)=-gt+c and from the initial conditions, r'(t)=U

Now, I guess that the constant c=Usinθ because that will give me t=2Usinθ/g when y=0, but I have no idea why?

I think the notation if the position vector is confusing me, and I just can't seem to figure out how to get the equation for r from r''=-gj, which i need for the second part of the question which is to find the displacement of the particle from the origin of the coordinate system.

Any pointers would be great!
Thanks

 

[hamsterman]

This is hardly calculus. The position vector of the particle is $(0+U\cos \theta * t)\vec{i} + (0+U\sin \theta * t - \frac{g*t^2}{2})\vec{j}$. That's a parabola. You just need to find its roots.