Question

A rail gun is a device like a train on a track, with the train propelled by a powerful electrical pulse. Very high speeds have been demonstrated in test models, and rail guns have been proposed as an alternative to rockets for sending into outer space any object that would be strong enough to survive the extreme accelerations. Suppose that the rail gun capsule is launched straight up, and that the force of air friction acting on it is given by F=be−ᶜˣ

, where x is the altitude, b and c are constants, and e is the base of natural logarithms. The exponential decay occurs because the atmosphere gets thinner with increasing altitude. (In reality, the force would probably drop off even faster than an exponential, because the capsule would be slowing down somewhat.) Find the amount of kinetic energy lost by the capsule due to air friction between when it is launched and when it is completely beyond the atmosphere. (Gravity is negligible, since the air friction force is much greater than the gravitational force.)

Answer 1

The kinetic energy lost by the railgun capsule due to air friction is found by integrating the force of air friction over the altitude it acts upon, from the ground to the edge of the atmosphere.

Step-by-step explanation:

To find the amount of kinetic energy lost by the capsule due to air friction from launch until it leaves the atmosphere, we can integrate the force of friction over the distance it acts, which in this case is altitude (x). Given that the force of friction is F = be^-cx, we can express the work done by friction (and thus the kinetic energy lost) as W = ∫ F dx = ∫ be^-cx dx. We must set the limits of integration from the launch altitude (usually ground level, so x = 0) to the edge of the atmosphere, which will be some finite altitude where air friction is no longer appreciable or the projectile is considered to be in space. The integration yields -b/c*e-cx evaluated from 0 to the edge of the atmosphere. We substitute the limits of integration to find the total kinetic energy lost to air friction.