physics problem??

Question

To attain the maximum range a projectile has to be launched at 45 degrees if the landing spot and the launch spot are at the same height (neglecting air resistance effects.)
Explain in a few sentences how the relation between the vertical and the horizontal components of the initial velocity affects the projectile range?

Answer 1

Okay, after scratching out some equations and looking at all this, here is my view.
First of all when you fire a projectile it will have an initial vertical velocity of vy and a horizontal velocity of vx.

The distance downrange achieved depends on how long the projectile can stay in the air which is dependent on the initial vy.

The time the projectile is in the air is simply T=vy/g where g is the graviational acceleration and vy is the initial v in the y direction.

the projectile will therefore travel D=vx*T downrange.

Substituting in for T we get D = vx*vy/g

Now, vx=v*cos(theta) and vy=v*sin(theta) where v is the initial velocity and theta is the angle.

So D = v*cos(theta)*v*sin(theta)/g

If we choose theta = 0 then sin = 0 and cos=1 so we get D=0

If we choose theta =90 then sin=1 and cos=0 so D=0

So we need something in between 0 and 90 that maximizes D. We can use some differentiation and actually calculate a maximum. OR we can consider that as we change the angle, sin gets bigger and cos gets smaller. But we are multiplying them together to get D. So we want to keep moving theta until their values are equal. that is at 45 degrees.

In short there is a trade off between time in the air and distance down range. More than 45 degrees and we have more time in the air but we have robbed the shot of some of its horizontal velocity. Less than 45 degrees and we have less time in the air, and the additional horizontal velocity is not enough to compensate.

Answer 2

In a few sentences. if the projectile is launched at 45 degrees, it reaches it's max range because the vertical and horizontal component of the initial velocity are equal to each other. And if it was launched at another angle then the vertical and horizontal components would be different therefore giving you a shorter range.

Answer 3

The more vertical velocity it has, the longer it stays up.

The longer it stays up, the more time it has for the horizontal velocity to carry it down range.

Answer 4

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