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How can we determine the distance a steel ball inside a grenade will fly horizontally taking into account air resistance?
If we assume that the detonation velocity is 6900m/s(TNT), the ball has a diameter of 5mm and the amount of tot. explosive is 180g.
Will the shrapnel immediately reach terminal velocity and how does the detonation pressure (0.86GPa) affect the terminal velocity?

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Terminal velocity is the maximum velocity attainable by an object as it falls through a fluid or in this case air. "Falling" in this case means moving vertically with a constant velocity when air resistance is equal to the force of gravity so terminal velocity would not be applicable here.

If you assume the ball is moving horizontally with velocity of 6900 m/s however, without air resistance you can calculate the maximum distance using a ballistic trajectory. With air resistance you have to take into account multiple components of the drag which will make the travel distance shorter than only using ballistic trajectory.

At subsonic velocities (below 330 m/s at sea level) main components of the drag will be form drag which you can calculate knowing that the form of a ball is sphere; and skin friction which is dependent of the material - in this case steel. At transonic and supersonic velocities you also have to take into account wave drag which occurs due to the presence of shock waves. As the aforementioned speed is almost 21 mach at standard atmospheric pressure, i assume the incoming air (relative to the ball) might also get ionised due to kinetic energy being sufficient and you might encounter extra energy loss due to thermal radiation.

To solve this problem you will need to take all these components into account, but remember that that the total force of drag will be in the opposite direction of the velocity.

Also the detonation pressure will dissipate with time as the air expands and will mostly affect the speed of sound.

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