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<title>At what angle to the vertical should a smooth chute be directed from point A so that the ball slides down it to the inclined plane in the shortest time?</title>
$1.3.3.$ At what angle to the vertical should a smooth chute be directed from point $A$ so that the ball slides down it to the inclined plane in the shortest time?
</p>
<center>
<figure>
<img src="statement.png"
loading="lazy" width="230" />
<figcaption>
For problem $1.3.3$
</figcaption>
</figure>
</center>
<p>
</p>
<h3>Solution</h3>
<p>
Let's consider the possible directions of the gutter from point $A$. It is clear that by changing the angle of the gutter to the plane, the length of the gutter changes, but the acceleration with which the ball will move along the gutter also changes.
From geometric considerations:
$$ h = x \cdot\tan{\alpha} + x \cdot\tan{\varphi}$$
$$ l = \frac{x}{\cos{\alpha}}\; (1) $$
The ball will roll with acceleration:
$a = g \cdot\sin{\alpha}$.
We find the time from the condition that the ball has traveled $l$:
<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="At what angle to the vertical should a smooth chute be directed from point A so that the ball slides down it to the inclined plane in the shortest time?">
<meta name="description" content="At what angle to the vertical should a smooth chute be directed from point A so that the ball slides down it to the inclined plane in the shortest time?">
<meta property="og:title" content="At what angle to the vertical should a smooth chute be directed from point A so that the ball slides down it to the inclined plane in the shortest time?">
<meta property="og:title" content="At what angle to the vertical should a smooth chute be directed from point A so that the ball slides down it to the inclined plane in the shortest time?">
<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="At what angle to the vertical should a smooth chute be directed from point A so that the ball slides down it to the inclined plane in the shortest time?">
<meta property="og:description" content="At what angle to the vertical should a smooth chute be directed from point A so that the ball slides down it to the inclined plane in the shortest time?">
<title>At what angle to the vertical should a smooth chute be directed from point A so that the ball slides down it to the inclined plane in the shortest time?</title>
<title>At what angle to the vertical should a smooth chute be directed from point A so that the ball slides down it to the inclined plane in the shortest time?</title>
$1.3.3.$ At what angle to the vertical should a smooth chute be directed from point $A$ so that the ball slides down it to the inclined plane in the shortest time?
$1.3.3.$ At what angle to the vertical should a smooth chute be directed from point $A$ so that the ball slides down it to the inclined plane in the shortest time?
</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 $1.3.3$
For problem $1.3.3$
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<p>
<p>
</p>
</p>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
Let's consider the possible directions of the gutter from point $A$. It is clear that by changing the angle of the gutter to the plane, the length of the gutter changes, but the acceleration with which the ball will move along the gutter also changes.
Let's consider the possible directions of the gutter from point $A$. It is clear that by changing the angle of the gutter to the plane, the length of the gutter changes, but the acceleration with which the ball will move along the gutter also changes.
From geometric considerations:
From geometric considerations:
$$ h = x \cdot\tan{\alpha} + x \cdot\tan{\varphi}$$
$$ h = x \cdot\tan{\alpha} + x \cdot\tan{\varphi}$$
$$ l = \frac{x}{\cos{\alpha}}\; (1) $$
$$ l = \frac{x}{\cos{\alpha}}\; (1) $$
The ball will roll with acceleration:
The ball will roll with acceleration:
$a = g \cdot\sin{\alpha}$.
$a = g \cdot\sin{\alpha}$.
We find the time from the condition that the ball has traveled $l$:
We find the time from the condition that the ball has traveled $l$:
<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>
