I was searching over previous posts about the Coriolis effect and saw people wondering how it works. I thought I would start a new thread and try to explain it a bit and how it applies to ballistics.
When we do physics of bullets, we use Newton's Law of Motion: F=ma or a=F/m. We find all of the forces that act on the bullet (F) and divide by the mass of the bullet (m) and get the acceleration of the bullet (a). We then integrate the acceleration to find the velocity and position of the bullet at any time.
Here is the problem, F=ma only works in what we call an "inertial" reference frame. That means that it is not accelerating. However, anyone that has driven around a curve knows that just going around a curve makes you feel acceleration. If the Earth is spinning about it's axis and the Earth is rotating around the sun and the solar system is spinning around the Milky Way, how the hell can we use it? Why can't we "feel" these accelerations?
The answer can be found in something called relative motion. If I want to use F=ma on Earth I have to correct it for these other motions or ignore them. When you apply these corrections you get three terms that pop out: Euler Acceleration, Centrifugal Acceleration, and Coriolis Acceleration. The math is what is called vector math but I can simplify it down to get rough estimates.
Euler Acceleration is based upon "angular acceleration" of how fast the spin is changing. Since the rotation of the Earth/Sun/Galaxy is pretty constant we can ignore it for ballistics.
Centrifugal Acceleration is a maximum of the spin squared times the radius. So for the Earth spins at 1 rotation per day with an average radius of 6378 km which gives us a centrifugal acceleration of 0.034 m/s^2 or about 0.003 "g" of acceleration. That is so small that it is not really noticeable in ballistics.
Coriolis though is based upon the speed of the object. It is 2 times the spin times the velocity. For throwing a ball it is not noticeable and for most engineering applications we can ignore Coriolis as well but with rifle bullets we start to get significant errors. The faster the bullet, the greater the error.
If you do the same math for Spin around the Sun/Galaxy you will see that they are insignificant.
So how do you correct for Coriolis? Well, it depends where you are on the Earth. The Earth spin vector points up from the North Pole. Part of it pushes the bullet to the right north of the Equator or to the left south of it. The other part of it either increases or decreases the apparent mass of bullet depending on if you are shooting east or west. This is the part that makes us launch rockets to the east as close to the equator as possible to reduce the apparent mass of the launch.
Given the uncertainty in calculating drop with drag, we often ignore the east/west apparent mass error but for long shots we need to include the small cross-range motion of the bullet. Most ballistic calculators include the cross range Coriolis correction. How do you know if yours does? If you need to enter in the Line of Fire (LOF) or direction that you are shooting and your latitude then your ballistic calculator can make the correction.
I hope this helps!
John Stutz, PhD
Aerospike Bullets, LLC
When we do physics of bullets, we use Newton's Law of Motion: F=ma or a=F/m. We find all of the forces that act on the bullet (F) and divide by the mass of the bullet (m) and get the acceleration of the bullet (a). We then integrate the acceleration to find the velocity and position of the bullet at any time.
Here is the problem, F=ma only works in what we call an "inertial" reference frame. That means that it is not accelerating. However, anyone that has driven around a curve knows that just going around a curve makes you feel acceleration. If the Earth is spinning about it's axis and the Earth is rotating around the sun and the solar system is spinning around the Milky Way, how the hell can we use it? Why can't we "feel" these accelerations?
The answer can be found in something called relative motion. If I want to use F=ma on Earth I have to correct it for these other motions or ignore them. When you apply these corrections you get three terms that pop out: Euler Acceleration, Centrifugal Acceleration, and Coriolis Acceleration. The math is what is called vector math but I can simplify it down to get rough estimates.
Euler Acceleration is based upon "angular acceleration" of how fast the spin is changing. Since the rotation of the Earth/Sun/Galaxy is pretty constant we can ignore it for ballistics.
Centrifugal Acceleration is a maximum of the spin squared times the radius. So for the Earth spins at 1 rotation per day with an average radius of 6378 km which gives us a centrifugal acceleration of 0.034 m/s^2 or about 0.003 "g" of acceleration. That is so small that it is not really noticeable in ballistics.
Coriolis though is based upon the speed of the object. It is 2 times the spin times the velocity. For throwing a ball it is not noticeable and for most engineering applications we can ignore Coriolis as well but with rifle bullets we start to get significant errors. The faster the bullet, the greater the error.
If you do the same math for Spin around the Sun/Galaxy you will see that they are insignificant.
So how do you correct for Coriolis? Well, it depends where you are on the Earth. The Earth spin vector points up from the North Pole. Part of it pushes the bullet to the right north of the Equator or to the left south of it. The other part of it either increases or decreases the apparent mass of bullet depending on if you are shooting east or west. This is the part that makes us launch rockets to the east as close to the equator as possible to reduce the apparent mass of the launch.
Given the uncertainty in calculating drop with drag, we often ignore the east/west apparent mass error but for long shots we need to include the small cross-range motion of the bullet. Most ballistic calculators include the cross range Coriolis correction. How do you know if yours does? If you need to enter in the Line of Fire (LOF) or direction that you are shooting and your latitude then your ballistic calculator can make the correction.
I hope this helps!
John Stutz, PhD
Aerospike Bullets, LLC
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