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

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

$$F(t) = \left \{ \begin{matrix} 0 & \mbox{ t

(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?

## closed as off-topic by Kyle Kanos, Aaron Stevens, Bob D, stafusa, ZeroTheHeroOct 13 at 15:23

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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

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

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 $$t and $$t>t_0+\delta t \mapsto t_0$$. What will the matching conditions imply?