Possible angles and ranges of a projectile

Question

A projectile is fired with initial speed 25m/s and passes through a point 10m up and 20m horizontally from the point of projection. Find the two possible angles of projection and for these two trajectories find the difference between their maximum heights and the difference between their ranges....... Tried drawing out and using trajectory equations but struggling to get anywhere and would really appreciate some help. Thank you

Answer 1

You have a system with two unknowns: $\theta$, the angle at which the projectile is fired, and $t$, the time it reaches $x=20$ and $y=10$. You have two equations: $$x=v_0\cos(\theta)t\\y=v_0\sin(\theta)t-gt^2/2$$ We move the terms that don't contain trig functions to one side $$x=v_0\cos(\theta)t\\y+gt^2/2=v_0\sin(\theta)t$$ Now square the equations and add them $$x^2+y^2+2ygt^2/2+(g/2)^2t^4=v_0^2t^2$$ If you plug in your numbers, this is a quadratic equation in $t^2$. You get two solutions for $t^2$, both positive in this case, corresponding to the two trajectories. Just take the square root for each, plug those numbers in the first equation, and you get two values for $\theta$. From there, you just apply formulas for maximum height and maximum range