# Projectile motion maximization

1. Feb 9, 2016

### fog37

Hello Forum,

I am aware that in order to maximize the time of flight, for an object being launched from a certain level and returning to the same level, the launching angle must be theta=90 degrees.
To maximize range instead (same level to same level) the angle should be theta=45 degrees (no air resistance).

What if we wanted to maximize both time of flight and range simultaneously? What approach would we choose? How would we manipulate the kinematic equations? The angle should be more than 45 (intuitively). But maybe the 45 degree angle which produces the longest distance requires the horizontal velocity component to act for the longest time possible.... So maybe 45 degrees maximizes both range and time of flight at the same time....

thanks,
fog37

2. Feb 9, 2016

### BvU

Hi fog,

You need to establish a weight function to optimize two variables using only one independent variable.
In other words: what is the relative importance of range versus duration ?

3. Feb 9, 2016

### gjonesy

I am thinking projectile design, perhaps a (lifting body design) like the old wingless Xplanes. But how would you fire it? Sabo maybe?

https://en.wikipedia.org/wiki/Lifting_body

4. Feb 9, 2016

### fog37

Thanks BvU.

That is an interesting comment. Could you give me a simple example about the weight function?

thanks,
fog37

5. Feb 9, 2016

### BvU

Well, if they pay you ten dollars for each second and two dollars for each meter, you want to maximize 10 * time-of-flight + 2 * distance

6. Feb 11, 2016

### CWatters

This is like painting a wall as red as possible and as green as possible at the same time. Can't be done unless you allow some combination of both. You need another rule/equation to work out how much of each colour to use.

7. Feb 11, 2016

### CWatters

Now I think about it....

If you had to achieve a specific range (that is less than the maximum possible) there are two possible launch angles. One has a longer flight time than the other. That's the only other way I can make sense of your question without introducing a weighting factor.

8. Feb 11, 2016

### gjonesy

When I referred to (lifting body) I was thinking in simple terms of shape of projectile and areo dynamics. Maximizing time of flight. That is if you are also building a launcher. I have no idea if it's possible to fire a projectile that once it loses its peek volosity if it will (glide) in an arc.

9. Feb 12, 2016

### CWatters

Sure is. I have a catapult launched glider that folds up for vertical launch and unfolds when it slows down..

10. Feb 12, 2016

### fog37

Hello,
thanks for all your comments. I just used Excel to make some calculation.

An object is launched from ground level and lands at ground level. The initial speed is 10 m/s. I have calculated the time of flight and range for different launching angles. The maximum range is obtained for theta=45 degrees. The maximum time is obtain for theta=90 degrees.

I then calculated the product (range* time of flight) for every different launching angle between 0 and 90. It turns out that the launching angle theta= 55 degrees gives the highest product between the calculated ranges and respective times of flight... Would that be the angle that manages to get the largest range and largest time possible at the same time?

Am I use a weighting factor, implicitly, when I look for the maximum product (range* time of flight)?

thanks,
fog37

11. Feb 12, 2016

### gjonesy

I was thinking more along the lines of a solid lifting body shaped object fired from a pneumatic cannon with a sabo. But I suppose the principal would be the same though. Thanks for the confirmation.

12. Feb 12, 2016