<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="A dog of mass m is tied with a leash of length L to a sledge of mass M > m. At the initial moment, the dog is next to the sled. What is the greatest distance a dog can move the sledge in one jerk, if the coefficients of friction of the dog's paws and the sled runners on the horizontal surface are the same?">
<meta property="og:title" content="A dog of mass m is tied with a leash of length L to a sledge of mass M > m. At the initial moment, the dog is next to the sled. What is the greatest distance a dog can move the sledge in one jerk, if the coefficients of friction of the dog's paws and the sled runners on the horizontal surface are the same?">
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@@ -14,9 +14,9 @@
<meta property="og:description" content="A dog of mass m is tied with a leash of length L to a sledge of mass M > m. At the initial moment, the dog is next to the sled. What is the greatest distance a dog can move the sledge in one jerk, if the coefficients of friction of the dog's paws and the sled runners on the horizontal surface are the same?">
<title>A dog of mass m is tied with a leash of length L to a sledge of mass M > m. At the initial moment, the dog is next to the sled. What is the greatest distance a dog can move the sledge in one jerk, if the coefficients of friction of the dog's paws and the sled runners on the horizontal surface are the same?</title>
$2.4.20^*.$ A dog of mass m is tied with a leash of length $L$ to a sledge of mass $M > m$. At the initial moment, the dog is next to the sled. What is the greatest distance a dog can move the sledge in one jerk, if the coefficients of friction of the dog's paws and the sled runners on the horizontal surface are the same?
</p>
<center>
<figure>
<img src="statement.png"
loading="lazy" width="180" />
<figcaption>
For problem $2.4.20^*$
</figcaption>
</figure>
</center>
<p>
</p>
<h3>Solution</h3>
<p>
The dog pulls the sled, performing work equal to
\[
A = \int_0^L F \, ds
\]
Where:
\[
F = \mu mg
\]
So we get:
\[
A = \mu mg L
\]
According to the work-energy theorem, we know that:
\[
A = \Delta K = \frac{mv^2}{2}
\]
After the dog has run a distance $L$, the leash tethered to the sled will stretch. This scenario can be analyzed as an inelastic collision.
\[
mv = (M + m)u
\]
In this scenario, the work done against the frictional force represents the energy lost by the system, which results in the deceleration of the sled-dog system. This loss of kinetic energy can be quantified using the work-energy principle:
<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="A dog of mass m is tied with a leash of length L to a sledge of mass M > m. At the initial moment, the dog is next to the sled. What is the greatest distance a dog can move the sledge in one jerk, if the coefficients of friction of the dog's paws and the sled runners on the horizontal surface are the same?">
<meta name="description" content="A dog of mass m is tied with a leash of length L to a sledge of mass M > m. At the initial moment, the dog is next to the sled. What is the greatest distance a dog can move the sledge in one jerk, if the coefficients of friction of the dog's paws and the sled runners on the horizontal surface are the same?">
<meta property="og:title" content="A dog of mass m is tied with a leash of length L to a sledge of mass M > m. At the initial moment, the dog is next to the sled. What is the greatest distance a dog can move the sledge in one jerk, if the coefficients of friction of the dog's paws and the sled runners on the horizontal surface are the same?">
<meta property="og:title" content="A dog of mass m is tied with a leash of length L to a sledge of mass M > m. At the initial moment, the dog is next to the sled. What is the greatest distance a dog can move the sledge in one jerk, if the coefficients of friction of the dog's paws and the sled runners on the horizontal surface are the same?">
<meta property="og:image" content="img/logo.png">
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@@ -14,9 +14,9 @@
<meta property="og:description" content="A dog of mass m is tied with a leash of length L to a sledge of mass M > m. At the initial moment, the dog is next to the sled. What is the greatest distance a dog can move the sledge in one jerk, if the coefficients of friction of the dog's paws and the sled runners on the horizontal surface are the same?">
<meta property="og:description" content="A dog of mass m is tied with a leash of length L to a sledge of mass M > m. At the initial moment, the dog is next to the sled. What is the greatest distance a dog can move the sledge in one jerk, if the coefficients of friction of the dog's paws and the sled runners on the horizontal surface are the same?">
<title>A dog of mass m is tied with a leash of length L to a sledge of mass M > m. At the initial moment, the dog is next to the sled. What is the greatest distance a dog can move the sledge in one jerk, if the coefficients of friction of the dog's paws and the sled runners on the horizontal surface are the same?</title>
<title>A dog of mass m is tied with a leash of length L to a sledge of mass M > m. At the initial moment, the dog is next to the sled. What is the greatest distance a dog can move the sledge in one jerk, if the coefficients of friction of the dog's paws and the sled runners on the horizontal surface are the same?</title>
$2.4.20^*.$ A dog of mass m is tied with a leash of length $L$ to a sledge of mass $M > m$. At the initial moment, the dog is next to the sled. What is the greatest distance a dog can move the sledge in one jerk, if the coefficients of friction of the dog's paws and the sled runners on the horizontal surface are the same?
$2.4.20^*.$ A dog of mass m is tied with a leash of length $L$ to a sledge of mass $M > m$. At the initial moment, the dog is next to the sled. What is the greatest distance a dog can move the sledge in one jerk, if the coefficients of friction of the dog's paws and the sled runners on the horizontal surface are the same?
</p>
</p>
<center>
<center>
<figure>
<figure>
<img src="statement.png"
<img src="statement.png"
loading="lazy" width="180" />
loading="lazy" width="180" />
<figcaption>
<figcaption>
For problem $2.4.20^*$
For problem $2.4.20^*$
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<p>
<p>
</p>
</p>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
The dog pulls the sled, performing work equal to
The dog pulls the sled, performing work equal to
\[
\[
A = \int_0^L F \, ds
A = \int_0^L F \, ds
\]
\]
Where:
Where:
\[
\[
F = \mu mg
F = \mu mg
\]
\]
So we get:
So we get:
\[
\[
A = \mu mg L
A = \mu mg L
\]
\]
According to the work-energy theorem, we know that:
According to the work-energy theorem, we know that:
\[
\[
A = \Delta K = \frac{mv^2}{2}
A = \Delta K = \frac{mv^2}{2}
\]
\]
After the dog has run a distance $L$, the leash tethered to the sled will stretch. This scenario can be analyzed as an inelastic collision.
After the dog has run a distance $L$, the leash tethered to the sled will stretch. This scenario can be analyzed as an inelastic collision.
\[
\[
mv = (M + m)u
mv = (M + m)u
\]
\]
In this scenario, the work done against the frictional force represents the energy lost by the system, which results in the deceleration of the sled-dog system. This loss of kinetic energy can be quantified using the work-energy principle:
In this scenario, the work done against the frictional force represents the energy lost by the system, which results in the deceleration of the sled-dog system. This loss of kinetic energy can be quantified using the work-energy principle:
<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>
