1.2 Kinematics

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# Engineering Physics - Fall 2024

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<h3 style="color: dodgerblue;">Mechanics   <span style="color: black;font-size:50%;">$\mu\eta\chi\alpha\nu\iota\kappa\eta~\tau\epsilon\chi\nu\eta~=~$mode of action</span> </h3>

<p style="color: tomato;">Kinematics   <span style="color: black;font-size:50%;">$\kappa\iota\nu\eta\mu\alpha~=~$motion</span>   
<span style="text-align: center;">description of motion disregarding <strong>forces.</strong></span>
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<p style="color: tomato;">Dynamics   <span style="color: black;font-size:50%;">$\delta\upsilon\nu\alpha\mu\iota\zeta~=~$force</span>  
<span style="text-align: center;">motion of <strong>bodies</strong> under the action of <strong>forces.</strong></span>
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<h3 style="color: black;font-size:90%;">$\kappa\iota\nu\eta\mu\alpha$:</h3>

measure the location (1D) of a <strong>point</strong> at various time instances:

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<h3 style="color: dodgerblue;">Functional representation of motion (1D point)</h3>

<p style="text-align: center;"><span>
$s(t)$  yields a distance for any t.</span>  
<span style="color: red;font-size:50%;">What form would you infer for the <i>free-fall</i> example?</span>
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A more diverse motion:

<strong>Speed</strong> = rate of change at a particular time intant.<br>
<span style="color: red;font-size:50%;">How much farther would we fall at a given height, if time progresses just a tiny bit `$\epsilon$`? <button onclick="https://lti.klicker.uzh.ch?redirectTo=https://pwa.klicker.uzh.ch/course/f172190f-4c31-46da-a526-221572432c2d/liveQuizzes">Live Quiz</button></span>

$$v(t)=\lim_{\epsilon\to 0}\frac{s(t+\epsilon)-s(t)}{\epsilon}$$

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<h4 style="color: dodgerblue;">Speed = time derivative of the position</h4>
<span style="color: red;font-size:50%;">In general, a derivative keeps track of a quantity which changes when a parameter (not necessarily time) changes <i>infinitesimally</i>.</span>

<div style="color: gray;text-align: center;margin-top: 30px;font-size: 50%;o">
$s, u, v, w$: functions of $t$; <br> $a, b, c, n$: arbitrary constants
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<table class="tabular">
    <th>Function</th>
    <th>Derivative</th>
<tr>
    <td class="r t li2">$s=t^n$</td>
    <td class="t">$\quad\frac{ds}{dt}=nt^{n-1}$</td></tr>
<tr>
    <td class="r li2">$s=cu$</td>
    <td>$\quad\frac{ds}{dt}=c\,\frac{du}{dt}$</td></tr>
<tr>
    <td class="r li2">$s=u+v+w+\dotsb\quad$</td>
    <td>$\quad\frac{ds}{dt}=\frac{du}{dt}+\frac{dv}{dt}+\frac{dw}{dt}+\dotsb$</td></tr>
<tr>
    <td class="r li2">$s=c$</td>
    <td>$\quad\frac{ds}{dt}=0$</td></tr>
<tr>
    <td class="b r li2">$s=u^av^bw^c\dotsm$</td>
    <td class="b">$\quad\frac{ds}{dt}=s\biggl(\!\frac{a}{u}\frac{du}{dt}\!+\!
    \frac{b}{v}\frac{dv}{dt}\!+\!\frac{c}{w}\frac{dw}{dt}\!+\!\dotsb\!\biggr)$</td></tr></tbody>
</table>

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<h4 style="color: dodgerblue;">Acceleration <span style="color:black;font-size:80%;"> = time derivative of the velocity = 2nd time derivative of the position</span> </h4>

$$a(t)=\frac{d}{dt}v(t)=\frac{d}{dt}\frac{d}{dt}s(t)\equiv \frac{d^2}{dt^2}s(t)$$

As an operation, `$\frac{d}{dt}$` has an inverse. <span style="color: red;font-size:50%;">Given the speed at some time, where will I be a short increment later?</span>

`\[s(t+\epsilon)=v(t)\cdot\epsilon~\Rightarrow~s(t_f)=\sum_n^N v(t_i+n\epsilon)\epsilon~\to~\int_{t_i}^{t_f}v(t)\,dt\]`

<span style="color: red;font-size:50%;">for the finite difference `$t_f-t_i=N\epsilon$` and the integral sign `$\int$` representing the infinite sum in the limit `$\epsilon\to0,~N\to\infty$` s.t. `$N\cdot\epsilon~=~$`constant.</span>

<p style="color: darkgreen;">Example in 2 dimensions:</p>

- <span style="color:black;font-size:60%;"> `$x:$` start from rest at `$x_0$` and move with constant speed `$v$`  hence:   `$x(t)=x_0+v\cdot t$`</span>
- <span style="color:black;font-size:60%;"> `$y:$` start from rest at `$y_0$` and move with constant acceleration `$g$`  hence:   `$y(t)=y_0+\frac{a}{2}\cdot t^2$`</span>  <span style="color: red;font-size:50%;">(why is this so?)</span>

`\[s\to\vec{s}=\begin{pmatrix}x\\y\end{pmatrix}\qquad\text{and}\qquad\frac{ds}{dt}\to\frac{d}{dt}\vec{s}=\begin{pmatrix}\frac{dx}{dt}\\\frac{dy}{dt}\end{pmatrix}\]`

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Parametric graph:

<div class="center"><strong style="color: red;">Important:</strong> How far does the point travel in a small time interval $\Delta t$?</div>

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  <div class="grid-item"><img src="/images/2D-motion.svgz"  style="display: block;margin-right: auto;max-height: 150px;"></div>
  <div class="grid-item" style="font-size: 14px;">$\Delta t$ <i>small</i> enough to assume that $\vec{v}=~$constant.<br>
We obtain the velocity at $t$ by taking $\Delta t\to 0$:
$$\lim_{\Delta t\to 0}\frac{\Delta s}{\Delta t}=\sqrt{\left(\frac{\Delta x}{\Delta t}\right)^2+\left(\frac{\Delta y}{\Delta t}\right)^2}=\sqrt{v_x^2+v_y^2}=\vert\vec{v}\vert$$
<button onclick="https://pwa.klicker.uzh.ch/course/f172190f-4c31-46da-a526-221572432c2d/join?pin=416049891">Live Quiz</button>
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