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| <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> | | <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> |
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| <p class="author"> | | <p class="author"> |
| Solutions of Savchenko Problems in Physics <br> | | Solutions of Savchenko Problems in Physics <br> |
| <i><b>knowledge must be free</b></i> | | <i><b>knowledge must be free</b></i> |
| </p> | | </p> |
| </header> | | </header> |
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| <h3 id="back-link"><a href="../../#1.4">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.4">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.4.11.$ The body is dropped over the slab at a height $h$ from it. The slab moves vertically upwards at a speed of $u$. Determine the time between two consecutive impacts of the body on the slab. The blows are absolutely elastic. | | $1.4.11.$ The body is dropped over the slab at a height $h$ from it. The slab moves vertically upwards at a speed of $u$. Determine the time between two consecutive impacts of the body on the slab. The blows are absolutely elastic. |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <p>Let's move to the reference frame associated with the plate. At the initial moment of time, the relative velocity is $u$ and is directed downwards.</p> | | <p>Let's move to the reference frame associated with the plate. At the initial moment of time, the relative velocity is $u$ and is directed downwards.</p> |
| <p>From the law of conservation of energy, we find the velocity of the body $v$ at the moment before the impact</p> | | <p>From the law of conservation of energy, we find the velocity of the body $v$ at the moment before the impact</p> |
| <p class="exp"> | | <p class="exp"> |
| $$ \frac{mv^2}{2} = \frac{mu^2}{2} + mgh $$ | | $$ \frac{mv^2}{2} = \frac{mu^2}{2} + mgh $$ |
| </p> | | </p> |
| <p>Where is the desired velocity </p> | | <p>Where is the desired velocity </p> |
| <p class="exp">$$v = \sqrt{u^2+2gh}$$</p> | | <p class="exp">$$v = \sqrt{u^2+2gh}$$</p> |
| <p>In this inertial reference frame, after an elastic rebound, the relative velocity will only change direction.</p> | | <p>In this inertial reference frame, after an elastic rebound, the relative velocity will only change direction.</p> |
| <p>A body thrown vertically will land after time</p> | | <p>A body thrown vertically will land after time</p> |
| <p class="exp"> | | <p class="exp"> |
| $$ t = \frac{2v}{g} $$ | | $$ t = \frac{2v}{g} $$ |
| </p> | | </p> |
| <p>Time sought:</p> | | <p>Time sought:</p> |
| <p class="exp"> | | <p class="exp"> |
| $$ \fbox{$t = \frac{2\sqrt{u^2 + 2gh}}{g}$} $$ | | $$ \fbox{$t = \frac{2\sqrt{u^2 + 2gh}}{g}$} $$ |
| </p> | | </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$t = 2\sqrt{u^2/g^2 + 2h/g}$$ | | $$t = 2\sqrt{u^2/g^2 + 2h/g}$$ |
