In summary, in lower-division physics classes, air resistance is often ignored to simplify projectile motion calculations. However, at higher speeds, air resistance becomes proportional to the square of the velocity. The transition between these two behaviors depends on the object's geometry and the nature of the air flow around it. This can be determined using terms such as exterior ballistics, external ballistics, drag coefficient, and Reynolds number. Understanding fluid dynamics can also help make sense of this information.
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Steven_Scott
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In lower-division physics classes, air resistance is usually ignored to make the mathematics of projectile motion easier to understand.

When air resistance is included, it's often stated that at lower velocities, air resistance is proportional to the velocity of the object,

Fair ∝ kv

At higher speeds, air resistance becomes proportional to the square of the velocity,

Fair ∝ kv2

What I'm wanting to know is, how do we know at what speed air resistance is simply proportional to velocity and when does it become proportional to the square of the velocity?

Is there an easier way to determine when this "transition" takes place or does it depend on the geometry of the object you're working with?

 
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This depends on the nature of the air flow around the object and therefore also on its geometry.
 
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Some good search terms are exterior ballistics, external ballistics, drag coefficient vs Reynolds number. Lots of good information, although most of it would make more sense if you had taken a class in fluid dynamics. The Reynolds number is a non-dimensional number that incorporates fluid density, fluid viscosity, object size, and object velocity. At higher velocities, the Mach number becomes important.
 
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What is air resistance?

Air resistance, also known as drag, is the force exerted by air on an object as it moves through the air.

How does air resistance affect projectile motion?

Air resistance can slow down the speed of a projectile and change its trajectory, causing it to deviate from its expected path.

What factors can affect the amount of air resistance on a projectile?

The amount of air resistance on a projectile can be affected by the shape, size, and speed of the projectile, as well as the density and viscosity of the air it is moving through.

Can air resistance be ignored in projectile motion?

In most cases, air resistance cannot be ignored in projectile motion as it can significantly impact the motion of the projectile. However, in some cases, such as when the projectile is small and moving at high speeds, air resistance may have a negligible effect.

How can air resistance be minimized in projectile motion?

Air resistance can be minimized by reducing the surface area and increasing the speed of the projectile. This can be achieved through streamlining the shape of the projectile and using materials that are less affected by air resistance.

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