Translated 1.5.1-1.5.2;1.5.3-1.5.7

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+ <meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
+ <meta name="description" content="a. The velocity of point A of a rigid body is v and forms an angle 45^{\circ} with the direction of line AB. The velocity of point B of this body is u. Determine the projection of the velocity of point B on the direction AB. b. The velocities of points A and B of a rigid body are v. The velocity of point C, located in the plane of line AB and vector v, is u > v. Find the projection of the velocity of point C on the axis perpendicular to the specified plane.">
+ <meta name="author" content="Aliaksandr Melnichenka">
+ <meta name="date" content="2023-10" scheme="YYYY-MM">
+ <meta property="og:title" content="a. The velocity of point A of a rigid body is v and forms an angle 45^{\circ} with the direction of line AB. The velocity of point B of this body is u. Determine the projection of the velocity of point B on the direction AB. b. The velocities of points A and B of a rigid body are v. The velocity of point C, located in the plane of line AB and vector v, is u > v. Find the projection of the velocity of point C on the axis perpendicular to the specified plane.">
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+ <meta property="og:description" content="a. The velocity of point A of a rigid body is v and forms an angle 45^{\circ} with the direction of line AB. The velocity of point B of this body is u. Determine the projection of the velocity of point B on the direction AB. b. The velocities of points A and B of a rigid body are v. The velocity of point C, located in the plane of line AB and vector v, is u > v. Find the projection of the velocity of point C on the axis perpendicular to the specified plane.">
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+ <title>a. The velocity of point A of a rigid body is v and forms an angle 45^{\circ} with the direction of line AB. The velocity of point B of this body is u. Determine the projection of the velocity of point B on the direction AB. b. The velocities of points A and B of a rigid body are v. The velocity of point C, located in the plane of line AB and vector v, is u > v. Find the projection of the velocity of point C on the axis perpendicular to the specified plane.</title>
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+ <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="../../#1.5">$\leftarrow$Back</a></h3>
+
+ <h3> Statement </h3>
+ <p>
+ $1.5.7.$ $a. $ The velocity of point $A$ of a rigid body is $v$ and forms an angle $45^{\circ}$ with the direction of line $AB$. The velocity of point $B$ of this body is $u$. Determine the projection of the velocity of point $B$ on the direction $AB$.
+</p>
+<center>
+<figure>
+<img src="https://savchenkosolutions.com/1/1.5.7/statement1.png"
+loading="lazy" width="130" />
+<figcaption>
+For problem 1.5.7.a
+ </figcaption>
+ </figure>
+ </center>
+<p>
+b. The velocities of points $A$ and $B$ of a rigid body are $v$. The velocity of point $C$, located in the plane of line $AB$ and vector $v$, is $u > v$. Find the projection of the velocity of point $C$ on the axis perpendicular to the specified plane.
+</p>
+
+</p>
+<center>
+ <figure>
+ <img src="https://savchenkosolutions.com/1/1.5.7/statement2.png"
+ loading="lazy" width="150" />
+ <figcaption>
+ For problem $1.5.7.b$
+ </figcaption>
+ </figure>
+</center>
+<p>
+ </p>
+
+ <h3>Solution</h3>
+ <p>
+ <p>a) If the distances between two points are constant, then if we project the velocity of point $A$ onto line $AB$, then the projection of the velocity of point $B$ onto line $AB$ is exactly the same</p>
+<p class="exp">$$\fbox{$u_{AB} = v/\sqrt{2}$}$$</p>
+
+<p>b) The velocities of $A$ and $B$ are the same because either the body does not rotate, or its rotation occurs around line $AB$. </p>
+<p>The fact that the velocity of point $C$ has a different value indicates that its velocity $\vec u$ is made up of the velocity $\vec v$ in the plane of the figure and $\vec u_1$ perpendicular to the plane.</a>
+<p class="exp">$$\vec{u}_1+\vec{v} = \vec{u}$$</p>
+<p>From where</a>
+<p class="exp">$$\fbox{$u_1 = \sqrt{u^2 - v^2}$}$$</p>
+ </p>
+
+ <h4>Answer</h4>
+ <p>
+ a. $u_{AB} = v/\sqrt{2}$<br>
+b. $u_1 = \sqrt{u^2 − v^2}$
+ </p>
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