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<title>It is raining heavily. Drop rate u. The ball slides on the asphalt at a speed of v. How many times in the same period of time does it get more drops than the same, but stationary ball? Does the answer change if the ball is not round?</title>
$1.4.17.$ It is raining heavily. Drop rate $u$. The ball slides on the asphalt at a speed of $v$. How many times in the same period of time does it get more drops than the same, but stationary ball? Does the answer change if the ball is not round?
</p>
<h3>Solution</h3>
<p>
<p>During time $t$, all the drops that are in a cylinder of height $ut$ and base area $S_1$ equal to the area of the central cross-section of the ball perpendicular to $u$ fall on a stationary ball. That is, in the first case</p>
<p class="exp">$$N_1 = utS_1$$</p>
<p>In the second case, we move to the reference frame of a rolling ball, then the drops fall on it with a speed of </p>
<p class="exp">$$u_1 = \sqrt{u^2 + v^2}$$</p>
<p class="exp">$$N_2 = u_1 t S_2$$</p>
<p>where $S_2$ is equal to the area of the central cross-section of the ball perpendicular to $u_1$. For a ball-ball $S_1 = S_2 = S$, and then</p>
<p>When the ball has a shape different from a sphere, its cross-sectional area will vary from the plane of the section </p>
<p class="exp">$$S_1\neq S_2\neq S$$</p>
<p>Thus, the answer obtained will also vary depending on the shape of the ball </p>
<p >NO: For additional reading, I recommend the discussion of this problem on <a href="https://ask.bc-pf.org/t/zadacha-pro-dozhd-i-myach-zadachnik-1001/3855" target="_blank">ask.bc-pf.org</a></p>
</p>
<h4>Answer</h4>
<p>
<p>More drops fall on a rolling ball in $\dfrac{N_2}{N_1} = \sqrt{1 + \dfrac{v^2}{u^2}}$ times.</p>
<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>
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<meta name="description" content="It is raining heavily. Drop rate u. The ball slides on the asphalt at a speed of v. How many times in the same period of time does it get more drops than the same, but stationary ball? Does the answer change if the ball is not round?">
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@@ -14,9 +14,9 @@
<meta property="og:description" content="It is raining heavily. Drop rate u. The ball slides on the asphalt at a speed of v. How many times in the same period of time does it get more drops than the same, but stationary ball? Does the answer change if the ball is not round?">
<meta property="og:description" content="It is raining heavily. Drop rate u. The ball slides on the asphalt at a speed of v. How many times in the same period of time does it get more drops than the same, but stationary ball? Does the answer change if the ball is not round?">
<title>It is raining heavily. Drop rate u. The ball slides on the asphalt at a speed of v. How many times in the same period of time does it get more drops than the same, but stationary ball? Does the answer change if the ball is not round?</title>
<title>It is raining heavily. Drop rate u. The ball slides on the asphalt at a speed of v. How many times in the same period of time does it get more drops than the same, but stationary ball? Does the answer change if the ball is not round?</title>
$1.4.17.$ It is raining heavily. Drop rate $u$. The ball slides on the asphalt at a speed of $v$. How many times in the same period of time does it get more drops than the same, but stationary ball? Does the answer change if the ball is not round?
$1.4.17.$ It is raining heavily. Drop rate $u$. The ball slides on the asphalt at a speed of $v$. How many times in the same period of time does it get more drops than the same, but stationary ball? Does the answer change if the ball is not round?
</p>
</p>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
<p>During time $t$, all the drops that are in a cylinder of height $ut$ and base area $S_1$ equal to the area of the central cross-section of the ball perpendicular to $u$ fall on a stationary ball. That is, in the first case</p>
<p>During time $t$, all the drops that are in a cylinder of height $ut$ and base area $S_1$ equal to the area of the central cross-section of the ball perpendicular to $u$ fall on a stationary ball. That is, in the first case</p>
<p class="exp">$$N_1 = utS_1$$</p>
<p class="exp">$$N_1 = utS_1$$</p>
<p>In the second case, we move to the reference frame of a rolling ball, then the drops fall on it with a speed of </p>
<p>In the second case, we move to the reference frame of a rolling ball, then the drops fall on it with a speed of </p>
<p class="exp">$$u_1 = \sqrt{u^2 + v^2}$$</p>
<p class="exp">$$u_1 = \sqrt{u^2 + v^2}$$</p>
<p class="exp">$$N_2 = u_1 t S_2$$</p>
<p class="exp">$$N_2 = u_1 t S_2$$</p>
<p>where $S_2$ is equal to the area of the central cross-section of the ball perpendicular to $u_1$. For a ball-ball $S_1 = S_2 = S$, and then</p>
<p>where $S_2$ is equal to the area of the central cross-section of the ball perpendicular to $u_1$. For a ball-ball $S_1 = S_2 = S$, and then</p>
<p>When the ball has a shape different from a sphere, its cross-sectional area will vary from the plane of the section </p>
<p>When the ball has a shape different from a sphere, its cross-sectional area will vary from the plane of the section </p>
<p class="exp">$$S_1\neq S_2\neq S$$</p>
<p class="exp">$$S_1\neq S_2\neq S$$</p>
<p>Thus, the answer obtained will also vary depending on the shape of the ball </p>
<p>Thus, the answer obtained will also vary depending on the shape of the ball </p>
<p >NO: For additional reading, I recommend the discussion of this problem on <a href="https://ask.bc-pf.org/t/zadacha-pro-dozhd-i-myach-zadachnik-1001/3855" target="_blank">ask.bc-pf.org</a></p>
<p >NO: For additional reading, I recommend the discussion of this problem on <a href="https://ask.bc-pf.org/t/zadacha-pro-dozhd-i-myach-zadachnik-1001/3855" target="_blank">ask.bc-pf.org</a></p>
</p>
</p>
<h4>Answer</h4>
<h4>Answer</h4>
<p>
<p>
<p>More drops fall on a rolling ball in $\dfrac{N_2}{N_1} = \sqrt{1 + \dfrac{v^2}{u^2}}$ times.</p>
<p>More drops fall on a rolling ball in $\dfrac{N_2}{N_1} = \sqrt{1 + \dfrac{v^2}{u^2}}$ times.</p>
<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>
