Translated 1.4.1-1.4.18
en/1.4.8.md
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| + | <meta name="description" content="In case of an elastic impact of a body against a fixed wall, its velocity v changes only in the direction. Determine the change after the impact of the velocity of this body, if the wall is moving: a) at a speed u towards the body; b) at a speed w < v in the direction of movement of the body."> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="In case of an elastic impact of a body against a fixed wall, its velocity v changes only in the direction. Determine the change after the impact of the velocity of this body, if the wall is moving: a) at a speed u towards the body; b) at a speed w < v in the direction of movement of the body."> | ||
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| + | <title>In case of an elastic impact of a body against a fixed wall, its velocity v changes only in the direction. Determine the change after the impact of the velocity of this body, if the wall is moving: a) at a speed u towards the body; b) at a speed w < v in the direction of movement of the body.</title> | ||
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| + | <h2>Solutions of Savchenko Problems in Physics</h2> | ||
| + | <p class="author"> | ||
| + | Aliaksandr Melnichenka <br/> | ||
| + | October 2023 | ||
| + | </p> | ||
| + | </header> | ||
| + | |||
| + | <h3 id="back-link"><a href="../../#1.4">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $1.4.8.$ In case of an elastic impact of a body against a fixed wall, its velocity $v$ changes only in the direction. Determine the change after the impact of the velocity of this body, if the wall is moving: a) at a speed $u$ towards the body; b) at a speed $w < v$ in the direction of movement of the body. | ||
| + | |||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/1/1.4.8/statement.png" | ||
| + | loading="lazy" width="180" /> | ||
| + | <figcaption> | ||
| + | For problem $1.4.8$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | <p>The whole point of the problem is to move to an inertial system associated with the wall, correctly describing the relative velocity.</p> | ||
| + | <p>$a)$ In the case when the wall "catches up" with the ball, it is necessary to move to a frame of reference in which the wall is at rest.</p> | ||
| + | <p>In this frame of reference, the relative velocity of the ball before the impact is</p> | ||
| + | <p class="exp">$$v_{0} + u$$</p> | ||
| + | <p>After the elastic impact, the velocity projection changes sign: it becomes equal to </p> | ||
| + | <p class="exp">$$(- v_{0} - u)$$</p> | ||
| + | <p>If we then return to the original (laboratory) frame of reference, where the wall moves with a velocity of $u$, then the velocity of the ball here is </p> | ||
| + | <p class="exp"> | ||
| + | $${v}' = - v_{0} - 2u$$</p> | ||
| + | <p>The change in velocity in this case is </p> | ||
| + | <p class="exp"> | ||
| + | $$\fbox{$\Delta v = {v}' - v_{0}$ $= - 2(v_{0} + u)$}$$ | ||
| + | </p> | ||
| + | <p>$b)$ Similarly, we need to go to the frame of reference in which the wall is at rest. </p> | ||
| + | <p>In this frame of reference, the relative velocity of the ball before the impact is $v_{0} - w$. After the elastic impact, the projection of the velocity changes sign: it becomes equal to </p> | ||
| + | <p class="exp"> | ||
| + | $$(- v_{0} + w)$$ | ||
| + | </p> | ||
| + | <p>If we then return to the original (laboratory) frame of reference, where the wall moves with the velocity $w$, then the velocity of the ball is </p> | ||
| + | <p class="exp"> | ||
| + | $${v}' = - v_{0} + 2w$$ | ||
| + | </p> | ||
| + | <p>The change in velocity in this case is </p> | ||
| + | <p class="exp"> | ||
| + | $$\fbox{$\Delta v = {v}' - v_{0}$ $= - 2(v_{0} - w)$}$$ | ||
| + | </p> | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $\begin{aligned}&{a) }\Delta v = -2(v + u).\quad б) \Delta v = -2(v - w). (\text{Projection on the direction of the initial}\\&\text{speeds are considered positive.)}\end{aligned}$ | ||
| + | </p> | ||
| + | |||
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| + | <br> | ||
| + | <p> | ||
| + | <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | ||
| + | </p> | ||
| + | <p> | ||
| + | <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> aliaksandr@savchenkosolutions.com <br></small> | ||
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| <meta name="description" content="In case of an elastic impact of a body against a fixed wall, its velocity v changes only in the direction. Determine the change after the impact of the velocity of this body, if the wall is moving: a) at a speed u towards the body; b) at a speed w < v in the direction of movement of the body."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
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| </head> | |||
| <body style=""> | |||
| <header style="text-align:center;"> | |||
| <h2>Solutions of Savchenko Problems in Physics</h2> | |||
| <p class="author"> | |||
| Aliaksandr Melnichenka <br/> | |||
| October 2023 | |||
| </p> | |||
| </header> | |||
| <h3 id="back-link"><a href="../../#1.4">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $1.4.8.$ In case of an elastic impact of a body against a fixed wall, its velocity $v$ changes only in the direction. Determine the change after the impact of the velocity of this body, if the wall is moving: a) at a speed $u$ towards the body; b) at a speed $w < v$ in the direction of movement of the body. | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/1/1.4.8/statement.png" | |||
| loading="lazy" width="180" /> | |||
| <figcaption> | |||
| For problem $1.4.8$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| <p>The whole point of the problem is to move to an inertial system associated with the wall, correctly describing the relative velocity.</p> | |||
| <p>$a)$ In the case when the wall "catches up" with the ball, it is necessary to move to a frame of reference in which the wall is at rest.</p> | |||
| <p>In this frame of reference, the relative velocity of the ball before the impact is</p> | |||
| <p class="exp">$$v_{0} + u$$</p> | |||
| <p>After the elastic impact, the velocity projection changes sign: it becomes equal to </p> | |||
| <p class="exp">$$(- v_{0} - u)$$</p> | |||
| <p>If we then return to the original (laboratory) frame of reference, where the wall moves with a velocity of $u$, then the velocity of the ball here is </p> | |||
| <p class="exp"> | |||
| $${v}' = - v_{0} - 2u$$</p> | |||
| <p>The change in velocity in this case is </p> | |||
| <p class="exp"> | |||
| $$\fbox{$\Delta v = {v}' - v_{0}$ $= - 2(v_{0} + u)$}$$ | |||
| </p> | |||
| <p>$b)$ Similarly, we need to go to the frame of reference in which the wall is at rest. </p> | |||
| <p>In this frame of reference, the relative velocity of the ball before the impact is $v_{0} - w$. After the elastic impact, the projection of the velocity changes sign: it becomes equal to </p> | |||
| <p class="exp"> | |||
| $$(- v_{0} + w)$$ | |||
| </p> | |||
| <p>If we then return to the original (laboratory) frame of reference, where the wall moves with the velocity $w$, then the velocity of the ball is </p> | |||
| <p class="exp"> | |||
| $${v}' = - v_{0} + 2w$$ | |||
| </p> | |||
| <p>The change in velocity in this case is </p> | |||
| <p class="exp"> | |||
| $$\fbox{$\Delta v = {v}' - v_{0}$ $= - 2(v_{0} - w)$}$$ | |||
| </p> | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $\begin{aligned}&{a) }\Delta v = -2(v + u).\quad б) \Delta v = -2(v - w). (\text{Projection on the direction of the initial}\\&\text{speeds are considered positive.)}\end{aligned}$ | |||
| </p> | |||
| <footer class="row container"> | |||
| <br> | |||
| <p> | |||
| <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | |||
| </p> | |||
| <p> | |||
| <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> aliaksandr@savchenkosolutions.com <br></small> | |||
| </p> | |||
| </footer> | |||
| </body> | |||
| </html> | |||