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+ <header style="text-align:center;">
+ <h2>Solutions of Savchenko Physics Textbook</h2>
+ <p class="author">
+ Aliaksandr Melnichenka <br/>
+ October 2023
+ </p>
+ </header>
+
+ <h3 id="back-link"><a href="../">$\leftarrow$Back</a></h3>
+
+ <h3> Statement </h3>
+ <p>
+ $1.1.8.$ The conveyor belt has speed $w$. Above the belt moves a machine, throwing $\nu$ balls per unit time. The balls stick to the belt. A ball counter with a photocell counts only the balls that have passed directly under it. How many balls will the counter count in a unit of time if the speed of the machine is $v < w$, the speed of the counter is $u < w$?
+<center>
+ <figure>
+ <img src="statement.png"
+ loading="lazy" width="230" />
+ <figcaption>
+ For problem 1.1.8
+ </figcaption>
+ </figure>
+</center>
+ </p>
+
+ <h3>Solution</h3>
+ <p>
+ <p>
+ 1. If the automaton and the counter are at rest, T.e. $v=u=0$, then the counter will register $N_0$, balls during the time $\Delta t$.
+ </p>
+ <p style="text-align: center;">
+ $$N_0 = \nu \Delta t$$
+ </p>
+ <p>
+ 2. If the automaton is at rest ($v = 0$) and the counter is moving with speed $u$, the number of registered particles is
+ </p>
+ <p style="text-align: center;">
+ $$N_1 = \nu \frac{w-u}{w+v} \Delta t$$
+ </p>
+ <p>
+ 3. When the automaton and the counter move, the number of particles during the time $\Delta t$ is defined as
+ </p>
+ <p style="text-align: center;">
+ $$N_2 = \nu \frac{w-u}{w-v} \Delta t$$
+ </p>
+ <p>
+ per unit of time
+ </p>
+ <p style="text-align: center;">
+ $$n = \nu \frac{w-u}{w-v}$$
+ </p>
+ </p>
+
+ <h4>Answer</h4>
+ <p>
+ $${ν}' = ν\frac{w − u}{w − v}$$
+ </p>
+
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