The response of a mass-spring system to an impulse function
As said above the Dirac delta function could serve as a mathematical model for an external force of large magntiude that acts for only a very short period of time..
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where
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Obviously no real function can satisfy beeing zero except at a single point and have an integral equal to one.
In Maple 
is expressed as the derivative of the Heaviside standard unit step function.
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It is instructive to use Maple to model such an instantaneous unit impulse by starting with the function
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)) = piecewise(`<`(t, a), 0, `<`(t, `+`(a, epsilon)), `/`(1, `*`(epsilon)), 0)](images/paper_151.gif) |
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A plot of the rectangular pulse for 
and 
gives
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The area of the rectangular pulse is equal to 1.
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where 
is Heavisides unit step function. We can compare the response of a damped mass-spring system to a rectangular pulse ))](images/paper_164.gif)
as 
with the response of the Dirac's delta function used by Maple.
The output of the Maple code impulse_func (written above) animates the response to the rectangular pulse ))](images/paper_166.gif)
by a linear second order differential equation with constant coefficients and with damping.
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Animation
The behavior of the response to a rectangular pulse by a linear second order differential equation with damping as 
compared to the response to Dirac's delta function.
The last animated frame shows little difference between the two responses when 
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The figure shows an animated response of the damped mass-spring system initially at rest. At time 
the system is suddenly given a sharp "hammerblow" modelled by 
)) = piecewise(`<`(t, a), 0, `<`(t, `+`(a, epsilon)), `/`(1, `*`(epsilon)), 0)](images/paper_152.gif)
