Projectile: Find range, given two displacement points

[rosemetal17]

Homework Statement

A particle is projected with initial velocity u\cos\alpha\vec{i}+u\sin\alpha\vec{j} m/s from a point 0 on a horizontal plane.

If this particle passes through two points whose displacements from 0 are 3\vec{i}+\vec{j} and \vec{i}+3\vec{j}

show that the range is \frac{13}{4} and that \tan\alpha = \frac{13}{3}

Homework Equations

Range

\frac{2u^2\sin\alpha\cos\alpha}{g}

S, displacement

S = ut+\frac{1}{2}at^2

The Attempt at a Solution

So, if the initial velocity is u\cos\alpha\vec{i}+u\sin\alpha\vec{j} m/s, then the particle is projected at u m/s at angle \alpha to the horizontal.

Then

S_x = ut\cos\alpha
S_y = ut\sin\alpha-\frac{gt^2}{2}

and if the displacement points are 3\vec{i}+\vec{j} and \vec{i}+3\vec{j}, then:

S_x = ut_1\cos\alpha = 3
S_y = ut_1\sin\alpha-\frac{gt_1^2}{2} = 1

and


S_x = ut_2\cos\alpha = 1
S_y = ut_2\sin\alpha-\frac{gt_2^2}{2} = 3

I tried solving for the last two pair of equation. For each pair, I eliminated t. So I got two separate equations in total involving \alpha and u only.

So here is where I'm stuck.
I tried solving for \alpha and u, but I just...can't!? I have no idea what I'm doing wrong, whether if it's algebraic error, or there's something wrong with the formulas. I've been stuck on this for the 5th hour now, and have redone this millions of times.. But still..

I'm beginning to suspect a problem with the question itself! Though I seriously doubt it..

Help'd be muchly appreciated!

Thanks in advance..

 

[ehild]

You are on the right track. Show your pair of equation for u and tan(alpha) to check. I they are right it is easy to eliminate the term containing u and solving for the angle.

ehild