because wiith the 45 degree angle we are going to have an optimal blend between horizontal and vertical velocities
According to your explanation then throwing it at a 30 degree angle would be better than throwing it at a 45 degree angle, which is just untrue. We want to look at both horizontal and vertical velocities because the vertical velocity will dictate time of flight.Don't know what combining horizontal and vertical velocities have to do with this, but...
Assuming distance in the x direction, then: d=(cos Vx)(t). Clearly from this equation there is a direct coorelation between the magnitude of vector Vx and d, therefore the higher values for cos should be the answer. If we think about this conceptually, throwing a pitch side arm almost parallel to the ground (for max value of cos=1 at angle of 0 degrees) would yield a further x distance than throwing it up in the air at a lesser angle--assuming equal velocities, of course.
I agree with IntelInside... a 45 degree angle is the optimal blend. Here's an example (slightly modified) taken from EK 1001because wiith the 45 degree angle we are going to have an optimal blend between horizontal and vertical velocities
Vf=Vo+a*tA 45 degree angle is clearly the magic angle that maximizes travel time and distance.
I agree with IntelInside... a 45 degree angle is the optimal blend. Here's an example (slightly modified) taken from EK 1001
A projectile is launched at an angle of 30 degrees to the horizontal. It's initial velocity is 20m/s. How far does it travel?
To solve this you have to first find total flight time. Using v=vo+at where v=0m/s at the top of the flight path; vo=vo(sin30); and a=-10m/s
We see that t=1 sec
That's only for half the trip though so total flight time is t=2 sec
Now to find total distance traveled we can use the equation d=vot+1/2at^2
Where a=0 because there's no acceleration in the x-direction; t=2 sec; and vo=vo(cos30)
We see that d=36m
Now, do the same problem with a 45 degree angle.
We see that t=3.4 sec and d=57.8m
Well, what about a 60 degree angle?
We see that t=3.6 sec and d=36m
A 45 degree angle is clearly the magic angle that maximizes travel time and distance.
If you use 30 degree, then projectile will land too soon comparing to 45 deg., which aka. less distance. You can also confirm the answer by setup a right triangle, and figure out what angles make the longest hypotenuse.
Here's the explanation in a nutshell: