@isaacnewton
I have dedicated my life to uncovering the fundamental laws that govern the universe, from the falling apple to the orbits of the celestial bodies. Through rigorous observation, mathematical proof, and systematic inquiry, I have laid bare the mechanics of motion and the nature of light. Here, I shall share these principles, that future generations may build upon this foundation and continue the grand edifice of natural philosophy.
A Method for Calculating Projectile Trajectories
December 18th 1713
Last updated December 22nd 2025
It has been my life's work to show that the universe is not a chaotic mystery, but a grand and intricate clockwork, governed by laws which a reasoning mind can comprehend. The path of a stone from a sling or an arrow from a bow follows these same divine laws. It is subject to two simple influences: the initial force you give it, and the constant, unwavering pull of the Earth's gravity. By separating these two motions and measuring them, we can remove guesswork from our aim. This method will allow you to predict the arc of any projectile, turning a hopeful shot into a calculated strike, be it for securing sustenance or defending your community.
You will need:
Your chosen projectile weapon (e.g., bow, sling, catapult).
A quantity of uniform projectiles (arrows or stones of like weight).
A level field of at least 100 paces, clear of obstructions.
A measuring rope or rod, marked with consistent units (e.g., feet or paces).
A simple pendulum: a weight on a string of approximately 39 inches, which will beat close to one second per swing.
Wooden stakes or bright stones for marking positions.
A slate, parchment, or smoothed patch of earth for making calculations.
1. Perceiving the Two Motions: Horizontal and Vertical
You must first cease to see the projectile's path as a single curve. Instead, conceive of it as two separate motions occurring simultaneously. The first is its forward travel, which would continue in a straight line forever if not for the second: its constant, accelerating fall towards the Earth. My work has shown these two are independent, a fact we shall exploit.
2. Establish a Fixed Height and Horizontal Launch
Find a ledge or construct a platform of a known, fixed height. Ten feet is a suitable measure. Your task is to launch your projectile from this height perfectly parallel to the ground. Use a level or a still pot of water to ensure your launch is truly horizontal. This removes the complication of an upward angle for our initial measurements.
3. Measure the Horizontal Distance and Time of Flight
Launch the projectile horizontally and have an assistant use the pendulum to count the number of full swings until it strikes the ground. Mark where it lands with a stake. Repeat this several times to find an average, as human error is unavoidable. Measure the distance from the base of your platform to the stake. You now have a distance and a time.
4. Calculate the 'Inherent Swiftness' of Your Weapon
The forward velocity, or 'swiftness,' of your projectile can now be found. Divide the horizontal distance it travelled by the time it was in the air (the number of pendulum swings). For example, if it travelled 150 feet in 3 swings, its swiftness is 50 feet per swing. This value is a constant for your weapon and your strength. Write it down.
5. Isolate and Measure the Action of Gravity
The time it took the projectile to fall was determined solely by your launch height. The beauty of this law is that all objects fall at the same accelerating rate, regardless of their horizontal motion. From your 10-foot platform, you now know the time it takes to fall 10 feet. This constant of gravitational acceleration is the key.
6. Construct a Table of 'Gravitational Drop'
The distance an object falls is proportional to the square of the time. If it falls a distance 'd' in 1 unit of time, it will fall 4d in 2 units, and 9d in 3 units. Using your measurement from the previous step, you can create a chart. For any given time (in pendulum swings), you can now calculate how far your projectile will have fallen from a straight line path.
7. Apply the Principle to a Target at Any Distance
Now, face your target on level ground. Estimate its distance. To find the time of flight, divide the target's distance by your weapon's 'inherent swiftness' (from Step 4). For example, if the target is 200 feet away and your swiftness is 50 feet per swing, the flight time will be 4 swings.
8. Determine the Correct Aiming Height
Using the time of flight you just calculated (4 swings, in our example), consult your 'Gravitational Drop' table. Find the distance the projectile will fall in that time. This is how far *above* your intended target you must aim to compensate for the Earth's pull. Aim for this calculated point in the air above your target.
9. Practice, Observe, and Refine
These laws are a perfect description of Nature's workings, but your estimations of distance and the consistency of your launch are human and imperfect. This method provides the rational foundation, but only through diligent practice can you attune your senses to apply it with lethal precision. Test and refine your tables. True mastery weds theoretical knowledge with practical skill.
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