09/06/2024
Using this image we can do an approximate calculation of the what speed the booster is traveling at the time of the photo.
The result is Mach 1.086
1339 km/h
832 mph
When a body is traveling higher than Mach 1 it produces a Mach wave. This is sometimes visible. If we can determine the angle of the Mach wave or sides of the triangle that make that angle, we can determine the Mach number of the object. Which is how many times faster than the speed of sound its traveling.
Once we have the Mach number, we can determine the velocity if we know the speed of sound at the location of the image.
Calculations for determining the velocity of an object that is producing a Mach wave:
Using the image (SpaceX booster producing a Mach wave) we measured an equivalent angle of 67.08 degrees, or a triangle that has longest side 38mm and y (opposite) side 35mm.
Note we will be using the triangle sides lengths to determine the result (not the angle as the side lengths is probably more accurate and we didn't have a protractor for measuring the angle on the image so we used the side lengths instead)
Formula for Mach angle to Mach number
Mach = 1 / sin(angle)
And
sin(angle) = y / r
So
Mach = 1 / (y / r) = r / y
where y and r are the opposite side and hypotenuse lengths.
Using inputs
Mach = 38 / 35 = 1.086
We will use 343 m/s for the speed of sound. The speed of sound can change depending on height and other factors, but we will use this value as the booster was still quite near the surface.
Velocity = Speed of sound x Mach
= 343 x 1.086
= 372 m/s
Or
1339 km/h
Or
832 mph
The v***r cone is amazing 👏 🤩
đź“·SpaceX