Question

A water balloon slingshot launches its projectiles essentially from ground level at a speed of 25.5 m/s . (You can ignore air resistance.)

A: At what angle should the slingshot be aimed to achieve its maximum range?

B: If shot at the angle you calculated in Part A, how far will a water balloon travel horizontally?

C: For how long will the balloon be in the air?

Answer 1

the formulae for range and flight time for a given launch from ground level at angle \(\theta \) are
\[ R = \frac{v^2 \sin 2 \theta} {g} \]
and
\[ T = \frac{2 v \sin \theta} {g} \]

these follow from the equations of motion and the parabolic/symmetric shape of flight without air resiatance

you can see that you will max out on range at \(\theta = 45^o\)

this covers the ground nicely
https://en.wikipedia.org/wiki/Range_of_a_projectile
and is the source of the equations i shared