To find out how high something can go and how far it can travel when launched into the air, we can use some simple formulas.

1. Maximum Height ($H$):

   You can calculate this using this formula:

   $H = \frac{v_0^2 \sin^2 \theta}{2g}$

   Here’s what the letters mean:

   • $v_0$ is how fast the object is thrown (initial velocity).
   • $\theta$ is the angle it’s thrown from (launch angle).
   • $g$ is the force of gravity, which is about $9.81 \, \text{m/s}^2$.

2. Range ($R$):

   To find out how far it travels, use this formula:

   $R = \frac{v_0^2 \sin(2\theta)}{g}$

Here’s an example:

If you throw something at a $30^\circ$ angle with a speed of $20 \, \text{m/s}$, just plug those numbers into the formulas to find out the maximum height ($H$) and the range ($R$).

Happy calculating!