Maximizing Projectile Distance: Calculation at 45 Degrees

[clipperdude21]

When you kick a ball of mass m into the air at a speed v0 at an angle theta to the ground, how do you know that the maximum distance in the x direction that the ball can travel occurs at 45 degrees.

Can someone show me the calculation of how this happens?

 

[andrevdh]

The formula for the range, $$R$$, of a projectile has a term

$$\sin(2\alpha)$$

where $$\alpha$$ is the launch angle above the horizon. This means that the range will be a maximum when the projectile is launched such that

$$2\alpha = 90^o$$

 

[clipperdude21]

thanks! how did u get that equation. is it possible to derive it from the kinematics equations? THanks!

nvm i think i did it