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| <h2>Solutions of Savchenko Physics Textbook</h2> | | <h2>Solutions of Savchenko Physics Textbook</h2> |
| <p class="author"> | | <p class="author"> |
| Aliaksandr Melnichenka <br/> | | Aliaksandr Melnichenka <br/> |
| October 2023 | | October 2023 |
| $1.1.8.$ 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 $u < w$? | | $1.1.8.$ 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 $u < w$? |
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| <figcaption> | | <figcaption> |
| For problem 1.1.8 | | For problem 1.1.8 |
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| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <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 $\Delta t$. | | 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> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $$N_0 = \nu \Delta t$$ | | $$N_0 = \nu \Delta t$$ |
| </p> | | </p> |
| <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 | | 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> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $$N_1 = \nu \frac{w-u}{w+v} \Delta t$$ | | $$N_1 = \nu \frac{w-u}{w+v} \Delta t$$ |
| </p> | | </p> |
| <p> | | <p> |
| 3. When the automaton and the counter move, the number of particles during the time $\Delta t$ is defined as | | 3. When the automaton and the counter move, the number of particles during the time $\Delta t$ is defined as |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $$N_2 = \nu \frac{w-u}{w-v} \Delta t$$ | | $$N_2 = \nu \frac{w-u}{w-v} \Delta t$$ |
| </p> | | </p> |
| <p> | | <p> |
| per unit of time | | per unit of time |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $$n = \nu \frac{w-u}{w-v}$$ | | $$n = \nu \frac{w-u}{w-v}$$ |
| </p> | | </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $${ν}' = ν\frac{w − u}{w − v}$$ | | $${ν}' = ν\frac{w − u}{w − v}$$ |
| </p> | | </p> |
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