Projectile motion, harder

[Leodip]

I was absent when my math teacher explained what would have been my favourite physics topic of this year, so I looked it up and studied it by myself. I'm fine with that topic and we already moved on, however there's a thing I'm still not sure of.
I showed a situatiin I invented to my teacher, and he said that without using math (and then sticking to physics stuff, mostly trigonometry) he couldn't find a way to solve it, even though he looked at it just for less than a minute.

Basically:
a: angle of shooting;
v: starting speed;
O: starting point of the projectile;
P: point pf maximum height;
Q: point of fall.
O and Q are situated on the x-axis.

Data:
OQ=OP

Calculate v and a (sticking to trigonometry)

I myself came up with a solution, but am not certain of it. Someone?

[dragtom]

without using that the trajectory follows a parabola I see no way to solve this.
Perhaps you'd say that that is allowed, but it is in fact physics.

[farscape]

Maybe I am misunderstanding something, or you made a typo, but if OQ-OP that would mean all of them are the same point, since O->P->Q would be a trict ordering, i.e. P would have to be between O and Q (unless there is a head-wind to blow back the projectile, but there is no info about that), so if OQ=OP that would mean P and Q falls on the same point on the x axis. If there is no horizontal movement after the highest point is reached, that could only happen if there wasn't any horizontal movement at all, so the projectile was going straight up and came down on the same spot, so a=90 degrees. This can happen with any value of 'v'.

[CuCN]

Maybe I am misunderstanding something, or you made a typo, but if OQ-OP that would mean all of them are the same point, since O->P->Q would be a trict ordering, i.e. P would have to be between O and Q (unless there is a head-wind to blow back the projectile, but there is no info about that), so if OQ=OP that would mean P and Q falls on the same point on the x axis. If there is no horizontal movement after the highest point is reached, that could only happen if there wasn't any horizontal movement at all, so the projectile was going straight up and came down on the same spot, so a=90 degrees. This can happen with any value of 'v'.

You're misunderstanding something: OPQ aren't in a straight line, since O and Q are on the ground and P is in the air.

[Leodip]

Basically, OPQ is an equilateral triangle. Given this, the angle is surely higher than 60 degrees, supposedly tg(a)= sqrt(12) which is 73.89788624801399, if I haven't miscalculated anything.

[CuCN]

Yes, tan a should be equal to 2sqrt(3), or sqrt(12).

If you let the projection of P onto the x-axis be R and the intersection of the original trajectory with line PR be S, PR=PS. (This is a property of parabolas and can be proved in several ways.) Because OPQ is equilateral, PR=OR*sqrt(3). tan a=RS/OR=2*PR/OR=2sqrt(3).

[Leodip]

Nice. Couldn't check my result, but would I be wrong saying that the speed the projectile is launched with doesn't change the fact that it'll form an equilateral triangle, as long as it is launched with that angle?

BTW, we haven't gone over parabolas, but still, we cannot use that, because it is math, indeed. I reached that conclusion noticing that OP=OQ and that OQ/2 and height max formed a triangle with OP. Doing calculations there led me to that conclusion.

[dragtom]

There are an infinite amount of formulae that go through O, P, and Q...

[CuCN]

Oh, I think I misread your original post as wanting to solve it without physics, instead of only physics.

The amount of time that the projectile takes to reach the maximum height is vsin(a)/g, and in this time it goes up by ((vsin(a))^2)/(2g) and horizontally by (v^2)sin(a)cos(a)/g. Since the vertical distance must be sqrt(3) times the horizontal distance, sin(a)/(2cos(a))=sqrt(3), so tan(a)=2sqrt(3).

[dragonsdemesne]

I'll try and remember to take a look at it when I get home.  I think it should be at least possible to solve it in terms of some sort of ratio between v and a or something, but maybe not with a definite number in mind.

[CuCN]

v and g do not matter at all, the answer only depends on a.

[Leodip]

v and g do not matter at all, the answer only depends on a.

Awesome. Thanks.