<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="Beads of mass m_1, m_2, m_3 can slide along the horizontal spoke without friction, with m_1 \gg m_2 and m_3 \gg m_2. Determine the maximum velocities of the extreme beads, if at first they were at rest, and the middle bead had a velocity v. The blows are elastic.">
<meta property="og:title" content="Beads of mass m_1, m_2, m_3 can slide along the horizontal spoke without friction, with m_1 \gg m_2 and m_3 \gg m_2. Determine the maximum velocities of the extreme beads, if at first they were at rest, and the middle bead had a velocity v. The blows are elastic.">
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@@ -14,9 +14,9 @@
<meta property="og:description" content="Beads of mass m_1, m_2, m_3 can slide along the horizontal spoke without friction, with m_1 \gg m_2 and m_3 \gg m_2. Determine the maximum velocities of the extreme beads, if at first they were at rest, and the middle bead had a velocity v. The blows are elastic.">
<title>Beads of mass m_1, m_2, m_3 can slide along the horizontal spoke without friction, with m_1 \gg m_2 and m_3 \gg m_2. Determine the maximum velocities of the extreme beads, if at first they were at rest, and the middle bead had a velocity v. The blows are elastic.</title>
$2.5.9^*.$ Beads of mass $m_1$, $m_2$, $m_3$ can slide along the horizontal spoke without friction, with $m_1 \gg m_2$ and $m_3 \gg m_2$. Determine the maximum velocities of the extreme beads, if at first they were at rest, and the middle bead had a velocity $v$. The blows are elastic.
</p>
<center>
<figure>
<img src="statement.png"
loading="lazy" width="230" />
<figcaption>
For problem $2.5.9^*$
</figcaption>
</figure>
</center>
<p>
</p>
<h3>Solution</h3>
<p>
Let's consider the velocity of the center of mass $v_c$:
$$
v_c = \frac{m_2 v}{m_1 + m_2 + m_3}
$$
According to the problem we know that $m_2 \ll m_1$ and $m_2 \ll m_3$, hence:
$$
v_c \approx 0
$$
Since the velocity of the center of mass is approximately zero, it follows that:
$$
m_1 v_1 = m_3 v_3 \quad (1)
$$
Now let's write down the law of conservation of energy, since the collisions are elastic:
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="Beads of mass m_1, m_2, m_3 can slide along the horizontal spoke without friction, with m_1 \gg m_2 and m_3 \gg m_2. Determine the maximum velocities of the extreme beads, if at first they were at rest, and the middle bead had a velocity v. The blows are elastic.">
<meta name="description" content="Beads of mass m_1, m_2, m_3 can slide along the horizontal spoke without friction, with m_1 \gg m_2 and m_3 \gg m_2. Determine the maximum velocities of the extreme beads, if at first they were at rest, and the middle bead had a velocity v. The blows are elastic.">
<meta property="og:title" content="Beads of mass m_1, m_2, m_3 can slide along the horizontal spoke without friction, with m_1 \gg m_2 and m_3 \gg m_2. Determine the maximum velocities of the extreme beads, if at first they were at rest, and the middle bead had a velocity v. The blows are elastic.">
<meta property="og:title" content="Beads of mass m_1, m_2, m_3 can slide along the horizontal spoke without friction, with m_1 \gg m_2 and m_3 \gg m_2. Determine the maximum velocities of the extreme beads, if at first they were at rest, and the middle bead had a velocity v. The blows are elastic.">
<meta property="og:image" content="img/logo.png">
<meta property="og:image" content="img/logo.png">
@@ -14,9 +14,9 @@
<meta property="og:description" content="Beads of mass m_1, m_2, m_3 can slide along the horizontal spoke without friction, with m_1 \gg m_2 and m_3 \gg m_2. Determine the maximum velocities of the extreme beads, if at first they were at rest, and the middle bead had a velocity v. The blows are elastic.">
<meta property="og:description" content="Beads of mass m_1, m_2, m_3 can slide along the horizontal spoke without friction, with m_1 \gg m_2 and m_3 \gg m_2. Determine the maximum velocities of the extreme beads, if at first they were at rest, and the middle bead had a velocity v. The blows are elastic.">
<title>Beads of mass m_1, m_2, m_3 can slide along the horizontal spoke without friction, with m_1 \gg m_2 and m_3 \gg m_2. Determine the maximum velocities of the extreme beads, if at first they were at rest, and the middle bead had a velocity v. The blows are elastic.</title>
<title>Beads of mass m_1, m_2, m_3 can slide along the horizontal spoke without friction, with m_1 \gg m_2 and m_3 \gg m_2. Determine the maximum velocities of the extreme beads, if at first they were at rest, and the middle bead had a velocity v. The blows are elastic.</title>
$2.5.9^*.$ Beads of mass $m_1$, $m_2$, $m_3$ can slide along the horizontal spoke without friction, with $m_1 \gg m_2$ and $m_3 \gg m_2$. Determine the maximum velocities of the extreme beads, if at first they were at rest, and the middle bead had a velocity $v$. The blows are elastic.
$2.5.9^*.$ Beads of mass $m_1$, $m_2$, $m_3$ can slide along the horizontal spoke without friction, with $m_1 \gg m_2$ and $m_3 \gg m_2$. Determine the maximum velocities of the extreme beads, if at first they were at rest, and the middle bead had a velocity $v$. The blows are elastic.
</p>
</p>
<center>
<center>
<figure>
<figure>
<img src="statement.png"
<img src="statement.png"
loading="lazy" width="230" />
loading="lazy" width="230" />
<figcaption>
<figcaption>
For problem $2.5.9^*$
For problem $2.5.9^*$
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<p>
<p>
</p>
</p>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
Let's consider the velocity of the center of mass $v_c$:
Let's consider the velocity of the center of mass $v_c$:
$$
$$
v_c = \frac{m_2 v}{m_1 + m_2 + m_3}
v_c = \frac{m_2 v}{m_1 + m_2 + m_3}
$$
$$
According to the problem we know that $m_2 \ll m_1$ and $m_2 \ll m_3$, hence:
According to the problem we know that $m_2 \ll m_1$ and $m_2 \ll m_3$, hence:
$$
$$
v_c \approx 0
v_c \approx 0
$$
$$
Since the velocity of the center of mass is approximately zero, it follows that:
Since the velocity of the center of mass is approximately zero, it follows that:
$$
$$
m_1 v_1 = m_3 v_3 \quad (1)
m_1 v_1 = m_3 v_3 \quad (1)
$$
$$
Now let's write down the law of conservation of energy, since the collisions are elastic:
Now let's write down the law of conservation of energy, since the collisions are elastic:
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
