# Question #ee158

##### 1 Answer

Starting at 0, increasing the angle up to

#### Explanation:

This does intuitively make sense.

Nonetheless, we can also prove this using a bit of algebra and calculus. Even if you can't do the second bit, first bit should make sense. The task is:

Consider this diagram.

Prove that the distance

Still here? Lets go.

Let's say we are firing a projectile with a velocity of

Considering vectors, the vertical velocity of the particle is

Find the time it takes for the particle to return to the ground if we throw it up and that velocity.

Using the equation

Now, consider the horizontal motion

since

Now, as an extra challenge, can you prove that the angle that gives the maximum r is

We can use some trig identities to help us with this expression, then differentiate to find the maximum angle.

Since

Now, differentiate both sides wrt

Since

Let

We want an angle between 0 and a right angle, so we want a value in the range