<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="Two trolleys of mass M each move in parallel with the initial speeds v_1 and v_2 ( ec{v}_2 > ec{v}_1). A load of mass m, which initially lay on the first trolley, is transferred to the second trolley with almost zero speed relative to this trolley. Then, with almost zero speed relative to the second trolley, it is transferred back to the first one. What will be the speed difference of the trolleys after N such transfers of cargo back and forth? Try to explain qualitatively the viscous friction that occurs when gas layers slip relative to each other.">
<meta property="og:title" content="Two trolleys of mass M each move in parallel with the initial speeds v_1 and v_2 ( ec{v}_2 > ec{v}_1). A load of mass m, which initially lay on the first trolley, is transferred to the second trolley with almost zero speed relative to this trolley. Then, with almost zero speed relative to the second trolley, it is transferred back to the first one. What will be the speed difference of the trolleys after N such transfers of cargo back and forth? Try to explain qualitatively the viscous friction that occurs when gas layers slip relative to each other.">
<meta property="og:image" content="img/logo.png">
@@ -14,9 +14,9 @@
<meta property="og:description" content="Two trolleys of mass M each move in parallel with the initial speeds v_1 and v_2 ( ec{v}_2 > ec{v}_1). A load of mass m, which initially lay on the first trolley, is transferred to the second trolley with almost zero speed relative to this trolley. Then, with almost zero speed relative to the second trolley, it is transferred back to the first one. What will be the speed difference of the trolleys after N such transfers of cargo back and forth? Try to explain qualitatively the viscous friction that occurs when gas layers slip relative to each other.">
<title>Two trolleys of mass M each move in parallel with the initial speeds v_1 and v_2 ( ec{v}_2 > ec{v}_1). A load of mass m, which initially lay on the first trolley, is transferred to the second trolley with almost zero speed relative to this trolley. Then, with almost zero speed relative to the second trolley, it is transferred back to the first one. What will be the speed difference of the trolleys after N such transfers of cargo back and forth? Try to explain qualitatively the viscous friction that occurs when gas layers slip relative to each other.</title>
$2.2.34.$ Two trolleys of mass $M$ each move in parallel with the initial speeds $v_1$ and $v_2$ ($\vec{v}_2 > \vec{v}_1$). A load of mass $m$, which initially lay on the first trolley, is transferred to the second trolley with almost zero speed relative to this trolley. Then, with almost zero speed relative to the second trolley, it is transferred back to the first one. What will be the speed difference of the trolleys after $N$ such transfers of cargo back and forth? Try to explain qualitatively the viscous friction that occurs when gas layers slip relative to each other.
</p>
<center>
<figure>
<img src="statement.png"
loading="lazy" width="250" />
<figcaption>
For problem $2.2.34$
</figcaption>
</figure>
</center>
<p>
</p>
<h3>Solution</h3>
<p>
From first to second:
$$m v_1 + M v_2 = (M + m) u_2$$
$$
u_2 = \frac{m v_1 + M v_2}{M + m}
$$
$$P_x = \text{const!}$$
Second to first:
$$
M v_1 + m u_2 = (M + m) u_1
$$
$$
u_1 = \frac{M v_1 + m u_2}{M + m}
$$
About the change:
$$
\Delta u = u_1 - u_2
$$
$$
u_1 > u_2
$$
$$
\Delta u = \left(\frac{M}{M + m}\right)^2 (v_1 - v_2)
<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="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="Two trolleys of mass M each move in parallel with the initial speeds v_1 and v_2 ( ec{v}_2 > ec{v}_1). A load of mass m, which initially lay on the first trolley, is transferred to the second trolley with almost zero speed relative to this trolley. Then, with almost zero speed relative to the second trolley, it is transferred back to the first one. What will be the speed difference of the trolleys after N such transfers of cargo back and forth? Try to explain qualitatively the viscous friction that occurs when gas layers slip relative to each other.">
<meta name="description" content="Two trolleys of mass M each move in parallel with the initial speeds v_1 and v_2 ( ec{v}_2 > ec{v}_1). A load of mass m, which initially lay on the first trolley, is transferred to the second trolley with almost zero speed relative to this trolley. Then, with almost zero speed relative to the second trolley, it is transferred back to the first one. What will be the speed difference of the trolleys after N such transfers of cargo back and forth? Try to explain qualitatively the viscous friction that occurs when gas layers slip relative to each other.">
<meta property="og:title" content="Two trolleys of mass M each move in parallel with the initial speeds v_1 and v_2 ( ec{v}_2 > ec{v}_1). A load of mass m, which initially lay on the first trolley, is transferred to the second trolley with almost zero speed relative to this trolley. Then, with almost zero speed relative to the second trolley, it is transferred back to the first one. What will be the speed difference of the trolleys after N such transfers of cargo back and forth? Try to explain qualitatively the viscous friction that occurs when gas layers slip relative to each other.">
<meta property="og:title" content="Two trolleys of mass M each move in parallel with the initial speeds v_1 and v_2 ( ec{v}_2 > ec{v}_1). A load of mass m, which initially lay on the first trolley, is transferred to the second trolley with almost zero speed relative to this trolley. Then, with almost zero speed relative to the second trolley, it is transferred back to the first one. What will be the speed difference of the trolleys after N such transfers of cargo back and forth? Try to explain qualitatively the viscous friction that occurs when gas layers slip relative to each other.">
<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="Two trolleys of mass M each move in parallel with the initial speeds v_1 and v_2 ( ec{v}_2 > ec{v}_1). A load of mass m, which initially lay on the first trolley, is transferred to the second trolley with almost zero speed relative to this trolley. Then, with almost zero speed relative to the second trolley, it is transferred back to the first one. What will be the speed difference of the trolleys after N such transfers of cargo back and forth? Try to explain qualitatively the viscous friction that occurs when gas layers slip relative to each other.">
<meta property="og:description" content="Two trolleys of mass M each move in parallel with the initial speeds v_1 and v_2 ( ec{v}_2 > ec{v}_1). A load of mass m, which initially lay on the first trolley, is transferred to the second trolley with almost zero speed relative to this trolley. Then, with almost zero speed relative to the second trolley, it is transferred back to the first one. What will be the speed difference of the trolleys after N such transfers of cargo back and forth? Try to explain qualitatively the viscous friction that occurs when gas layers slip relative to each other.">
<title>Two trolleys of mass M each move in parallel with the initial speeds v_1 and v_2 ( ec{v}_2 > ec{v}_1). A load of mass m, which initially lay on the first trolley, is transferred to the second trolley with almost zero speed relative to this trolley. Then, with almost zero speed relative to the second trolley, it is transferred back to the first one. What will be the speed difference of the trolleys after N such transfers of cargo back and forth? Try to explain qualitatively the viscous friction that occurs when gas layers slip relative to each other.</title>
<title>Two trolleys of mass M each move in parallel with the initial speeds v_1 and v_2 ( ec{v}_2 > ec{v}_1). A load of mass m, which initially lay on the first trolley, is transferred to the second trolley with almost zero speed relative to this trolley. Then, with almost zero speed relative to the second trolley, it is transferred back to the first one. What will be the speed difference of the trolleys after N such transfers of cargo back and forth? Try to explain qualitatively the viscous friction that occurs when gas layers slip relative to each other.</title>
$2.2.34.$ Two trolleys of mass $M$ each move in parallel with the initial speeds $v_1$ and $v_2$ ($\vec{v}_2 > \vec{v}_1$). A load of mass $m$, which initially lay on the first trolley, is transferred to the second trolley with almost zero speed relative to this trolley. Then, with almost zero speed relative to the second trolley, it is transferred back to the first one. What will be the speed difference of the trolleys after $N$ such transfers of cargo back and forth? Try to explain qualitatively the viscous friction that occurs when gas layers slip relative to each other.
$2.2.34.$ Two trolleys of mass $M$ each move in parallel with the initial speeds $v_1$ and $v_2$ ($\vec{v}_2 > \vec{v}_1$). A load of mass $m$, which initially lay on the first trolley, is transferred to the second trolley with almost zero speed relative to this trolley. Then, with almost zero speed relative to the second trolley, it is transferred back to the first one. What will be the speed difference of the trolleys after $N$ such transfers of cargo back and forth? Try to explain qualitatively the viscous friction that occurs when gas layers slip relative to each other.
</p>
</p>
<center>
<center>
<figure>
<figure>
<img src="statement.png"
<img src="statement.png"
loading="lazy" width="250" />
loading="lazy" width="250" />
<figcaption>
<figcaption>
For problem $2.2.34$
For problem $2.2.34$
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<p>
<p>
</p>
</p>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
From first to second:
From first to second:
$$m v_1 + M v_2 = (M + m) u_2$$
$$m v_1 + M v_2 = (M + m) u_2$$
$$
$$
u_2 = \frac{m v_1 + M v_2}{M + m}
u_2 = \frac{m v_1 + M v_2}{M + m}
$$
$$
$$P_x = \text{const!}$$
$$P_x = \text{const!}$$
Second to first:
Second to first:
$$
$$
M v_1 + m u_2 = (M + m) u_1
M v_1 + m u_2 = (M + m) u_1
$$
$$
$$
$$
u_1 = \frac{M v_1 + m u_2}{M + m}
u_1 = \frac{M v_1 + m u_2}{M + m}
$$
$$
About the change:
About the change:
$$
$$
\Delta u = u_1 - u_2
\Delta u = u_1 - u_2
$$
$$
$$
$$
u_1 > u_2
u_1 > u_2
$$
$$
$$
$$
\Delta u = \left(\frac{M}{M + m}\right)^2 (v_1 - v_2)
\Delta u = \left(\frac{M}{M + m}\right)^2 (v_1 - v_2)
<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>
