Thread: 45 degrees

I have learnt that in theory a projectile launched at 45 degrees will give the maximum range of that projectile in those conditions (eg. wind etc).

But what i fail to understand is why this is so? From experimentation, i have found that the horizontal and vertical velocity of a bottle rocket at 45 degrees is the same. Is the balance between the vertical and horizontal velocity the reason behind maximum range being attained by 45 degrees?

Thanks in advance.

It may have something to do with the stability of the projectile. Most rockets and shells are spinning in flight to give them more stability and therefore longer range. If the angle of flight is too steep you'll loose this spin (there will be a 'dead moment' at maximum hight), so 45 degrees could be the limit for a normal stable flight of a spinning projectile.

hasn't it lso got something to do with the arc of travel. if you fired it at 90degrees to ground, then all distance travelled will all be vertically, too low an angle and the arc is too shallow to make any great distance. 45degrees is half way between horizontal and vertical and would give the best distance without loosing as much on vertical travel

Thats what i thought anyway :-D

when shells were being shot into Israel they observers would figure angle shot and determine there destinations. at 45 degree they figured no particular destination and maximum distance the goal. this was also the way the recent No. Korean destinations or in this case no destinations were determined.

rockets or long range artillery have guidance systems. I'm not sure but i don't think a spin is considered. for the most part, shots into space are started at 90 degree or straight up and the angles they take prior to shutting off there mains are to conform to the objective orbit. if there is no intended orbit the direction has been pre planned and straight up the usual direction well past any earth gravity effects.

From the mathematical standpoint, there are two ways to look at it (off the top of my head... perhaps there are more ways...).

From the equations of projectile motion, we get that the horizontal distance is given by

x = (v₀)² sin(2 theta)/g.

To find the maxima, using calculus, we just take the derivative with respect to theta and find out where that equals zero;

dx/dtheta = 2(v₀)² cos(2 theta)/g = 0.

This equals zero when cos(2 theta) = 0.

This happens when the argument of cos is 90 (or 270...).

Solve 2*theta = 90 => theta = 45. (270 corresponds to firing the projectile at an angle of 135, i.e., in the reverse direction.)



A simpler way is to just ask where

x = (v₀)² sin(2 theta)/g

is at its maximum. Recall that -1 <,= sin <,= 1. That is, sin is always trapped between -1 & +1. So the expression for x is at its max when sin(2 theta) = +1, which happens at theta = 45.

Cheers,
william

I believe it works like this (this'll just be a basic look at it). As long as two projectiles are fired with the same force, they will both travel the same distance. However, one that is fired more vertically will use more of its distance travelling up rather than forward, and one fired more horizontally I think will be taken by gravity too quickly... Also, maybe someone other than I could elaborate on what you get when you calculate the vector of the projectile... Horizontal component vs. vertical. Will the combination of the vectors be greater if the components are equal rather than if one component was greater than the other? I think that's just pythagorean theorem basically...

Originally Posted by Chemboy

I believe it works like this (this'll just be a basic look at it). As long as two projectiles are fired with the same force, they will both travel the same distance. However, one that is fired more vertically will use more of its distance travelling up rather than forward, and one fired more horizontally I think will be taken by gravity too quickly... Also, maybe someone other than I could elaborate on what you get when you calculate the vector of the projectile... Horizontal component vs. vertical. Will the combination of the vectors be greater if the components are equal rather than if one component was greater than the other? I think that's just pythagorean theorem basically...

yeah thats what i was trying to say(badly :wink: )