Calculating energy from distance

If the projectile distance is known, it is possible to estimate a lower bound on projectile energy: (note: this assumes the slinger is on flat ground) $$E ≥ {1 \over 2} m g d$$ Where $g = 9.8 {m \over s^2}$.

If the launch angle is further known (for instance, by analyzing the frames of a video tape), then a more exact estimate can be obtained: $$E = {m g d \over 2 sin(2 \theta)}$$

Footnote

The projectile distance equation for flat ground is: $$d = {v_0^2 \over g} sin(2 \theta)$$

The maximum projectile distance occurs with a launch angle of 45°, so we can omit the launch angle by using an inequality: $$d ≤ {v_0^2 \over g} sin(2 \cdot 45 ^{\circ} )$$ $$d ≤ {v_0^2 \over g}$$

Solve for $v_0$: $$v_0 ≥ \sqrt {g d}$$

Plug that into the kinetic energy equation: $$E = {1 \over 2} m v_0^2$$ $$E ≥ {1 \over 2} m g d$$

An example query for Google Calculator is .5 * (9.8 m / s^2) * 100 grams * 100 meters.

Including launch angle

$$d = {v_0^2 \over g} sin(2 \theta)$$ $$v_0 = \sqrt{ g d \over sin(2 \theta) }$$ $$E = {1 \over 2} m v_0^2 = {m g d \over 2 sin(2 \theta)}$$

Records

Slinger	Date	Type	Mass	Throw style	Sling length	Range	Energy
Melvin Gaylor *	1970		212.6g			349.6m	≥364J
Vernon Morton *			283.5g			258.2m	≥358J
SEB		Stone	300g	Side-Arm	130cm	~220m	≥325J
Colonel Walker		Orange	~454g	Modified Underhand	122cm	~130m	≥290J
Douglas	02/11/05	Heavy stone	~500g			~90m	≥220J