Find $X$ and $U$ for projectiles.

Question

A projectile is fired from horizontal ground at an angle of $X$ above the horizontal and speed of $u$ m/s and lands a distance of 100m away. Another projectile is fired with the same initial speed but at an angle of $2X$ above the horizontal and lands a distance of 150m. What are $X$ and $u$? I have tried to use the projectiles equation I have, but I feel like it's not going well. Any assistance would be great, thanks.

Answer 1

The range of a projectile, launched from the horizontal, with speed $u$ at an angle $\theta$ is $$d=\frac{u^2}{g}\sin(2\theta)$$ You are given $d_1=100$m, for an angle $X$ and $d_2=150$m for an angle $2X$By dividing the equation for the two cases you get $$\frac{d_1}{d_2}=\frac{100}{150}=\frac{\sin(2X)}{\sin(4X)}$$ You can use $\sin(2a)=2\sin(a)\cos(a)$ to write $\sin(4a)=2\sin(2a)\cos(2a)$. Plug it into the above equation and you get a formula for $\cos(2X)$. Calculate $X$ and plug it into the first equation