(Kick Stop) Consider a mass on a spring where the extension of the spring $x(t)$ satisfies
$$m \ddot{x}+k x=I δ(t-T),
$$where $m$ is the mass, and $k>0$ is the spring constant. Suppose initially $x(0)=a$ and $\dot{x}(0)=0$ and that at time $t=T$ an instantaneous impulse $I$ is applied to the mass.
Obtain the motion of the mass for $t>0$, and find conditions on $I$ and $T$ such that the impulse completely stops the motion. Explain the result physically.
Practical Applied Mathematics(实用应用数学) chapter 9 page 119
In a very similar vein, recall the concept of an impulse in mechanics. In one-dimensional motion, the velocity $v$ of a particle under a force $f(t)$ satisfies Newton's equation
$$
m \frac{d v}{d t}=f(t)
$$
from which
$$
v(t)=v(0)+\frac{1}{m} \int_0^t f(s) d s .
$$
If the force is very large but only lasts for a short time, say
$$
f(t)= \begin{cases}I / \epsilon & 0<t<\epsilon \\ 0 & \text {otherwise}\end{cases}
$$
then we can integrate the equation of motion from $t=0$ to $t=\epsilon$ to find
$$
v(\epsilon)=\frac{1}{m} \int_0^\epsilon \frac{I}{\epsilon} d t=\frac{I}{m}
$$
Letting $\epsilon \rightarrow 0$, we have the result of an
impulse $I$: the velocity $v$ changes discontinuously from 0 to $I / m$. Again, we can ask the question, can we put the limiting impulse directly into the equation of motion, rather than having to smooth it out and take a limit?
Section 6: Dirac Delta Function
6. 1. Physical examples
Consider an 'impulse' which is a sudden increase in momentum $0 \rightarrow m v$ of an object applied at time $t_0$ say. To model this in terms of an applied force i.e. a 'kick' $F(t)$ we write
$$
m v=\int_{t_0-\tau}^{t_0+\tau} F(t) d t
$$
which is dimensionally correct, where $F(t)$ is strongly peaked about $t_0$ (see figure).
Actually the details of the shape of the peak are not important, what is important is the area under the curve. To obtain the limit of an impulse we would like to make the duration $(2 \tau)$ of the force smaller but the strength greater such that $m v$ remains the same. Then in the limit $\tau \rightarrow 0$ we obtain an impulse. In this limit we write
$$
F(t)=m v \delta\left(t-t_0\right)
$$
