<meta name="description" content="A website with solutions to physics problems from Savchenko Textbook">
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<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
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<meta name="description" content="The largest dataset of solutions of 'Savchenko. Problems in Physics'. Savchenko’s Problems in General Physics is widely used to prepare for olympiads and it is a useful tool to
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master and sharpen your skills and techniques in comptetitive problem solving. Some of these problems were a source
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of inspiration for Jaan Kalda’s handouts and to some NBPhO problems. You may find problems from old IPhO
$2.5.9^*.$ 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> astrosander01@gmail.com <br></small>
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="A website with solutions to physics problems from Savchenko Textbook">
<meta name="description" content="The largest dataset of solutions of 'Savchenko. Problems in Physics'. Savchenko’s Problems in General Physics is widely used to prepare for olympiads and it is a useful tool to
master and sharpen your skills and techniques in comptetitive problem solving. Some of these problems were a source
of inspiration for Jaan Kalda’s handouts and to some NBPhO problems. You may find problems from old IPhO
$2.5.9^*.$ 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> astrosander01@gmail.com <br></small>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small>
