An Army cannon is designed so that when a shell is fired from it, the carriage t

Question

An Army cannon is designed so that when a shell is fired from it, the carriage that the gun tube assembly sits on remains stationary, while the gun tube assembly slides in recoil. When a shell is fired, two components between the gun tube assembly and the carriage absorb the recoil: a damping mechanism and a recoil spring.

The recoil spring has the additional function of pushing the gun tube assembly back to the starting position, so that another shell can be loaded and fired.

The gun tube assembly has a mass of 1,600 kilograms.

The recoil spring has a spring constant of 20,000 Newtons per meter.

The damping mechanism exerts a force, in Newtons, numerically equal to 9,500 times the instantaneous velocity, in meters per second, of the gun tube assembly.

For this problem, assume that, when the cannon is fired, the gun tube assembly is pushed to the rear with an initial instantaneous velocity of 5 meters per second (to the right in the diagram). Assume that motion of the gun tube assembly to the right in the diagram is negative.

1.   Model this system as an initial-value differential equation problem, where

      x is the displacement of the gun tube assembly from the starting position (x = 0) and

      t is time in seconds.

Explanation / Answer

16(d^2x/dt^2)=95(dx/dt)+200x

You can solve it using the standard method to solve second order differential equations

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