- #1

Trying2Learn

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- TL;DR Summary
- Is this process, correct?

May I ask if the following process is correct?

Given: F=ma

Apply an impulsive force using the dirac delta near 0 (with F nearly constant over the tiny impulsive interval)

ma = Fδ(t)

This is a second order differential equation with a forcing function. However, I cannot readily integrate this differential equation.

Instead, I turn to Linear Momentum:

Initial momentum + (integral of force over time) = Final momentum

mv

With zero initial velocity, I now have:

v

And now I turn my original differential equation to this

ma = 0

With these two initial conditions:

x(0) = 0

v(0) = F/(mΔt)

The solution is:

x(t) = (F/(mΔt)) * t

This seems strange to me.

Is this process correct?

Can someone explain in words (sorry, I am embarrassed) what I am doing (if this is correct)?

Would I get the same results by solving the original equation with a convolution or numerical method?

It seems so strange to me: as if I skirted the complexity of a nonhomogeneous differential equation (I cheated).

Given: F=ma

Apply an impulsive force using the dirac delta near 0 (with F nearly constant over the tiny impulsive interval)

ma = Fδ(t)

This is a second order differential equation with a forcing function. However, I cannot readily integrate this differential equation.

Instead, I turn to Linear Momentum:

Initial momentum + (integral of force over time) = Final momentum

mv

^{+}=FΔt + mv^{-}With zero initial velocity, I now have:

v

^{+}= F/(mΔt)And now I turn my original differential equation to this

ma = 0

With these two initial conditions:

x(0) = 0

v(0) = F/(mΔt)

The solution is:

x(t) = (F/(mΔt)) * t

This seems strange to me.

Is this process correct?

Can someone explain in words (sorry, I am embarrassed) what I am doing (if this is correct)?

Would I get the same results by solving the original equation with a convolution or numerical method?

It seems so strange to me: as if I skirted the complexity of a nonhomogeneous differential equation (I cheated).