Technical post.
TLDR: Time difference analysis, combined with information on bullet velocities, gives a possible range of
120 to 161 meters for the distance of the shooter(s) from Trump. Additionally, depending on whatever distance in that range is used for the first shot, all other shots must have come either from the
same distance, or from a second rifle no more than 6-7 meters (20-23 feet) away than that distance (in the direction of Trump, not behind). With those ranges in mind,
all values are consistent with Crooks firing all 8 shots from a distance of approximately 141 meters from Trump using 5.56 ammunition fired from an AR-15.
The most important shot is the first one, for a few reasons: it was the one that hit Trump, it was the closest to the microphone, and it also hit David Dutch and/or the rail behind him and to his left, giving us the most accurate data for trajectory and distance. The graph below only shows calculations for the first shot only, and for rounds shot from a 16-inch barrel (see long version for details). It gives a range of 136-160 meters.
View attachment 100076
red curve: solution for average velocity and distance of the first shot (uncertainty due to speed of sound: +/-1 meter or so)
red line: minimum distance
green line: maximum distance
purple line: Crooks's distance
dotted orange line: fast bullet (55 grain)
dotted black line: slow bullet (77 grain)
Long version:
I
graphed the
equation for time difference between shock wave and muzzle blast using the largest (
red curve = 0.221 ms) and smallest (
green curve = 0.212 ms) time differences from the Trump mic recording, and overlaid the average
velocities of fast (
purple = Winchester Target 5.56x45 55 grain) and slow (
blue = Winchester Match 5.56x45 77 grain) brands of 55.6 ammunition, which is used in most AR-15s, at various distances. (FWIW, news reports claim Crooks bought
5.56 ammo, and that the
rifle was a
DPMS AR-15 5.56 bought in 2013, which chambers 223s and 5.56s.)
View attachment 100070
The time difference equation uses average velocity, not muzzle velocity or velocity at full distance, so when it solves for a particular distance, the bullet velocity is actually for a distance
half the value of the distance traveled. For example, a solved value of 200 m will give an average velocity equal to the velocity at 100 m. Therefore, I doubled the distance numbers given by the bullet manufacturers (velocity at 100 yards =
average velocity of a 200-yard shot). Since the manufacturer specs are usually for 20-inch barrels, the actual velocities will be
lower than shown if the barrel is 16 inches (the length of DPMS AR-15s) by something like 300 f/s or 91 m/s (
black = slow 5.56,
orange = fast 5.56 fired from 16-inch barrels).
I calculated 350.8 m/s as the speed of sound in Butler that day, given the temperature and elevation. But others have calculated it at plus or minus 1 m/s, so I used 350 for the left curve and 352 for the right one to give a wider range and account for that uncertainty.
This is rough, but it should give at least a fairly accurate approximation of the possible values, give or take a few meters. With the added proviso that I could have gotten something wrong: basically, the shooter(s) of all 8 bullets couldn’t have been closer than around 120 meters (if firing fast bullets from a 20-inch barrel, 917 m/s average velocity - vertical
blue line), or farther than around 161 meters (if firing slow bullets from a 16-inch barrel, 681 m/s average velocity - vertical
green line). 120 meters is about the distance to the outer wall of the AGR building. 161 meters is about halfway up the slope of the building behind Crooks. The best calculation I’ve found for how far Crooks was from Trump is 141 meters (155 yards) - verticle
purple line. (The circles are just roughly placed on the pic below to give an idea - I didn't try to get Trump's location exactly.)
View attachment 100072
For comparison, I eyeballed the possible ranges for other calibers of ammo used in semiautomatic rifles:
.300 Win Mag and .300 Blackout: 122-137m
.308 Win: 132-148m
.223 Rem: 110-140m
6.5 Creedmore: 127-148m
.243 Win (semiauto?): 110-134m
.458 SOCOM and .50 Beowulf: no match (all varieties too slow)
7.62: 165-180m
The combined possible range of all these different calibers is around 110-150 meters (again, we can probably add 10-20 meters to take into account shorter barrel lengths, giving us 110-170 meters). 110 is on the grass between the fence and the AGF, and 170 m is still on the second building, but on the farther slope. The outlier is 7.62, which would also place a shooter on the building behind Crooks, closer to the third building. Basically, the shots couldn't have come from anywhere behind the building behind Crooks, and probably came from the building Crooks was on.
A note on the time differences: The difference for the first shot is around 0.221 s, the second and eighth are 0.218 s, and the third and fourth are 0.212 s (the fifth through seventh shots are missing either shock waves or muzzle blasts in the recording, so can’t be calculated). There are several different factors that can cause such differences, including 1) shots fired from different guns at different distances (in which case all three first shots would come from different rifles, the farthest being around 6 or 7 meters behind the closest. Others online who have used the time difference equation have reached that conclusion (i.e. different shooters for each shot), but there are other factors, including: 2) a change in bullet trajectory from the same gun (the farther the trajectory from the mic, the shorter the time difference), 3) different grain amounts causing slightly differing velocities for individual bullets, and 4) the tendency of some rifles to lose velocity on each bullet as the barrel heats up. On the last point,
here’s an example of a guy whose velocities went down 20 to 30 ft/s for each bullet fired. With all that in mind, the differences between the shots are well within the variations those factors can cause - i.e. they do not rule out all shots fired from the same location.
Since no one heard bullets that sounded like they were being fired from inside a building, and the line of sight from the first floor windows was probably obscured by the risers and the people standing on them, it looks like all 8 shots were fired from Crooks’s position. Any second shooter would have had to have been basically sitting up there with him.
Some supporting analyses from the Peak Prosperity forum:
Greg_N performed a time difference of arrival analysis from 3 recordings which placed
all 8 shots within a 2m radius of Crooks’s location. (Interestingly, this analysis also shows both shots 9 and 10 coming from behind the north stands - in other words it doesn’t confirm the 10th shot coming from the south barn and seems to imply that both those shots came from the ground.)
VT used the Copenhaver video (and other measurements) to calculate the
distance/trajectory of the first bullet. It aligns with Crooks’s location and not the
windows below.
View attachment 100071
View attachment 100073
