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.

Solutions of Savchenko Problems in Physics

Aliaksandr Melnichenka
October 2023

    <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 $AB$.

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.

    <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>

$$\fbox{$u_{AB} = v/\sqrt{2}$}$$

b) The velocities of $A$ and $B$ are the same because either the body does not rotate, or its rotation occurs around line $AB$.

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.

$$\vec{u}_1+\vec{v} = \vec{u}$$

From where

$$\fbox{$u_1 = \sqrt{u^2 - v^2}$}$$

    <h4>Answer</h4>
    <p>
        a. $u_{AB} = v/\sqrt{2}$<br>

b. $u_1 = \sqrt{u^2 − v^2}$

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