External Ballistics: Elevation/ Declination

I have previously touched on external ballistics and drop, but as my focus is target shooting and not hunting, I had not made a study of elevation. In this blog entry, I discuss:

• Compensation for drop by aiming above the target
  
• How a target that is higher or lower than the shooter affects the flight of the projectile

Please note! When it comes to compensating for target range and elevation, I would say it's better to judge from experience than to use a table of numbers. Tables and charts multiply measurement and calculation inaccuracies, whereas the human eyes and brain are naturally very good at integrating all this information and predicting an aiming solution. However, by understanding the way that external ballistics works and is computed, we can better equip ourselves to visualise the projectile's flight, particularly in unusual situations of extreme elevation.

A Simplified Approach

External ballistics is an incredibly sophisticated and detailed science. Firearms and artillery shooters can compensate for so many factors that the wind direction may have changed or the target moved on by the time the final variable is measured and entered into the ballistics computer. Fortunately many of these factors become significant only at high velocities and for a typical slingshot with a dense projectile, just about the only significant force acting on the projectile is gravity as drag is immaterial compared to the momentum of the projectile. The only variables that we need to consider are:

• velocity of the projectile leaving the pouch;
  
• angle of elevation;
  
• distance to the target; and
  
• height of the target relative to the slingshot

Gravity

After you shoot a slingshot, gravity acts on the projectile. Gravity acts as if a force constantly accelerates the projectile faster and faster towards the earth at 9.81 metres per second for each second of flight time. The longer the ball remains in flight, the faster the projectile will fall towards the ground. Seeing as a round ball that is not spinning does not generate any lift, then within reasonable limits, it doesn't matter how big or heavy the projectile is, it will always have the same acceleration due to gravity of 9.81m/s/s. The flight time and hence the distance dropped is therefore down to the distance to target (range) and the projectile's speed (velocity).

The following chart demonstrates this:



Projectiles shot at targets that are further away see exponentially more drop. You can compensate by increasing velocity. If you can shoot twice as fast at a target twice as far away, the drop will be the same, as the flight time will be the same. For example, a projectile shot at 80m/s at a range of 80m drops the same distance as one shot at a velocity of 40m/s and at a range of 40m because they both travel for a total flight time of 1 second. However, it's not all that easy to vary the velocity exactly as we want it, so we instead must aim upwards and use some of the projectile's forward momentum to compensate for drop.


Elevation

At the short distances and reasonably high speeds typical of target shooting, not much drop is experienced. Also, the distance does not vary. If you're the sort of shooter that likes to aim the fork at the target, then the easiest way to compensate for drop is to measure how many fork widths above or below the target you must place the top edge of the slingshot and always use the same offset. If you stick to the same bandset at the same draw length firing the same weight of projectile and you hold the fork the same distance from your eye, the velocity will be the same so the elevation compensation will not change from shot to shot.


It gets a lot trickier when the target range is very far or you are flinging heavy rocks. This is often the situation of hunters whose quarry will not let them approach and who must shoot heavy rocks or ball. In this case they will find themselves arcing up into the air and down onto their target.

Objects thrown up in this way to fall under gravity follow a parabolic arc.

A Bouncing Basketball (Wikipedia)



The horizontal distance travelled by a projectile, the maximum height attained and the flight time depend on the angle of elevation and are easy to understand.

Effect of Angle of Elevation on Range, Maximum Height and Flight Time (Wikipedia)



High arcs give high heights and long flight times, but do not travel far. Low angles result in a flatter trajectory, but they soon hit the ground. If you wish to shoot as far as possible, you would shoot upwards at a 45 degree angle. This is not the longest lasting flight, but it offers the longest range.

These are described as:

R=(v_i²sin2Θ_i)/g
h=(v_i²sin²Θ_i)/2g


and

Θ=0.5sin-1(gR/v²)

where:

• R= max range
  
• h= max height
  
• v_i= initial velocity
  
• g= acceleration due to gravity
  
• Θ_i= initial angle of elevation from the horizontal

Note that there are two possible solutions for the angle of elevation. One is a shallow arc and one is a high arc. For example, in the chart above, you can see that both the 75 degree and 15 degree arcs land in the same place, as do the 30 degree and 60 degree arcs. A slingshot shooter would generally select the most shallow angle.

Don't fret about the math. I've made a chart and if you check the references section HyperPhysics has a very useful calculator for calculating the initial angle of arc.



I don't expect many people to actually use these charts and formulae to rain bullets down onto their targets, although they can be used to come up with a rough idea of how high to shoot. They are more important to illustrate general concepts, such as:

• For small elevations of under say 15 to 20 degrees, the ratio of elevation angle to range is almost linear.
  
• For angles above 30 degrees, much bigger increments of arc are needed to get a little more range.
  
• It gets tricky to hit the target using high elevations. At near 45 degrees, consider yourself lucky to reach.
  
• You can't reach a target if you can't get that far with a 45 degree elevation.
  
• Arcs above 45 degrees can be used to clear obstacles, but there'll be more vertical travel component than horizontal.
  
• Fast projectile-bandset combinations shot within ranges that one'd be accurate enough to hit a small target (say up to 100m) can do so with little drop and small angle changes
  
• Slow projectile-bandset combinations have significant limitations at long ranges. You cannot reach a target 100m away with a projectile travelling 30m/s.
  
• Above about 70% of the maximum range, estimating the required elevation becomes tricky.
  
• Any velocity produces a corresponding maximum range at a given angle.
  
• If you shoot a slingshot straight up, it will come straight back down. The speed it hits you on the noggin is the speed you shot it, less air resistance and never more than its terminal velocity.
  
• If you shoot your slingshot at say, 45 degrees and the slingshot is close to the ground on flat terrain, you can estimate its velocity based for far the bullet flies before it first lands.

That last comment sounds useful; we can estimate initial velocity without a chrony. Let's take a closer look.

R_max = v_i²/g

so

v_i = (gR_max)^-2

e.g. if the maximum range = 90m, then the initial velocity is the square root of 9.81m/s/s times 90m or ..... 29.7m/s

These are the velocities implied by a given range at a 45 degree arc.

41m=>20m/s
92m=>30m/s
163m=>40m/s
255m=>50m/s
367m=>60m/s
499m=>70m/s
652m=>80m/s
826m=>90m/s
1,019m=>100m/s

And here's a chart.



Yes, that implies that a projectile travelling a not-unheard-of speed like 100m/s has a potential range of over one kilometre. Of course, at these velocities and with light projectiles, air resistance is a factor, so you have to dial down your expectations a lot, but at the lower end of the scale, it should be indicative.

In my comment below, I will address some more advanced topics.

Blog References:
http://slingshotforu...-of-slingshots/
http://slingshotforu...timal-velocity/

External References:
http://hyperphysics....raj.html#tracon
http://www.pha.jhu.e...m/l5/node3.html
http://en.wikipedia....wiki/Trajectory
http://en.wikipedia....fleman%27s_rule
http://web.archive.o...ory_part_1.html