Find $v(t)$ and $x(t)$, How do I treat $δt$? [closed]

Question

We apply a force to a particle with a mass $m$ and inicial velocity $v_0$:

$$ F(t) = \left \{ \begin{matrix} 0 & \mbox{ $t<t_0$} \\ \frac{p_0}{\delta t} & \mbox{ $t_0<t<t_0 +\delta t$}\\ 0 & t_0+\delta t<t\end{matrix}\right. $$

(a) Find $v(t)$ and $x(t)$

(b) Show that the movement reach a constant velocity when $dt\rightarrow 0$ at $t=t_0$

My main question is:

How do I work with $\delta t$?

I suposed that $\delta t$ is a constant, so:

$$F(t)=\frac{p_0}{\delta t}=m\cdot a(t)$$

$$a(t)=\frac{dv}{dt}(t)=\frac{F(t)}{m}$$

Integrating: $$\int_{v_0}^{v}dv= \int_{t_0}^{t_0+\delta t}\frac{1}{m}F(t)dt=\frac{1}{m}\int_{t_0}^{t_0+\delta t}\frac{p_0}{\delta t} dt$$

$$ v-v_0=\frac{p_0}{m\delta t}\left[ t\right]_{t_0}^{t_0+\delta t}= \frac{p_0}{m\delta t}\left[ t_0+\delta t-t_0\right] $$

So

$$\fbox{$v(t)=v_0 + \frac{p_0}{m}$}$$

Knowing that $v(t)=dx(t)/dt$, we calculate in the same form $x(t)$ by integration, I obtain:

$$\fbox{$x(t)=x_0+v_0 \delta t + \frac{p_0}{m}\delta t$}$$

Is this correct?

How do I demostrate the second question?

Answer 1

You have to be more precise with the calculations. First consider the time interval $t < t_0$. There we have $\frac{dv}{dt} = 0$ and therefore $v(t)=v_0 = const.$ and

$x(t) =x_0+v_0t$.

A very similar result you obtain for the time interval $t > t_0 + \delta t$ (but different initial conditions!).

In the time interval in-between you have a constant acceleration; remind that $\int t dt = \frac{t^2}{2}$.

Now, to obtain the full $x(t),v(t)$, you have to match the motions for the three time intervals at their initial- and endpoints; you must ensure that the functions $x(t)$ and $v(t)$ are continuous in time. For example, a matching condition is

$x_{t_0<t<t_0+\delta t}(t=t_0)=x_{t<t_0}(t=t_0)$

where $x_{t<t_0}$ is the partial solution only regarding time interval $t<t_0$.

Now to b): The in-between time interval goes to Zero when $\delta t \mapsto 0$ and thus we are left with the two time intervals $t<t_0$ and $t>t_0+\delta t \mapsto t_0$. What will the matching conditions imply?