The conveyor belt has speed w. Above the belt moves a machine, throwing u 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? For problem 1.1.8

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Solutions of Savchenko Problems in Physics
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    <h3 id="back-link"><a href="/#1.1">$\leftarrow$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  $u < w$?

    <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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