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| <title>Inside a sphere of radius R moving at speed u, there is a ball of radius r, which at the moment when it passes through the center of the sphere, has a velocity v perpendicular to the velocity u. The mass of the sphere is much greater than the mass of the ball. Determine the frequency with which the ball hits the wall of the sphere. The blows are absolutely elastic.</title> | | <title>Inside a sphere of radius R moving at speed u, there is a ball of radius r, which at the moment when it passes through the center of the sphere, has a velocity v perpendicular to the velocity u. The mass of the sphere is much greater than the mass of the ball. Determine the frequency with which the ball hits the wall of the sphere. The blows are absolutely elastic.</title> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.4.10.$ Inside a sphere of radius $R$ moving at speed $u$, there is a ball of radius $r$, which at the moment when it passes through the center of the sphere, has a velocity $v$ perpendicular to the velocity $u$. The mass of the sphere is much greater than the mass of the ball. Determine the frequency with which the ball hits the wall of the sphere. The blows are absolutely elastic. | | $1.4.10.$ Inside a sphere of radius $R$ moving at speed $u$, there is a ball of radius $r$, which at the moment when it passes through the center of the sphere, has a velocity $v$ perpendicular to the velocity $u$. The mass of the sphere is much greater than the mass of the ball. Determine the frequency with which the ball hits the wall of the sphere. The blows are absolutely elastic. |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <p>Considering that the sphere is very massive, the speed of the ball in the inertial reference frame will be constant and equal to</p> | | <p>Considering that the sphere is very massive, the speed of the ball in the inertial reference frame will be constant and equal to</p> |
| $$ v_{rel} = \sqrt{v^2+u^2}$$ | | $$ v_{rel} = \sqrt{v^2+u^2}$$ |
| <p>Between two successive impacts, the center of the ball travels a path equal to </p> | | <p>Between two successive impacts, the center of the ball travels a path equal to </p> |
| <p class="exp">$$ l = 2(R-r)$$</p> | | <p class="exp">$$ l = 2(R-r)$$</p> |
| <p>Then we find the time interval $\Delta t$ between two successive collisions as</p> | | <p>Then we find the time interval $\Delta t$ between two successive collisions as</p> |
| <p class="exp"> | | <p class="exp"> |
| $$ \Delta t = \frac{l}{v_{rel}} = \frac{2(R-r)}{\sqrt{v^2+u^2}} $$ | | $$ \Delta t = \frac{l}{v_{rel}} = \frac{2(R-r)}{\sqrt{v^2+u^2}} $$ |
| </p> | | </p> |
| <p>Find the collision frequency</p> | | <p>Find the collision frequency</p> |
| <p class="exp"> | | <p class="exp"> |
| $$ \boxed{\nu = \Delta t ^{-1} = \frac{\sqrt{v^2+u^2}}{2(R-r)}} $$ | | $$ \boxed{\nu = \Delta t ^{-1} = \frac{\sqrt{v^2+u^2}}{2(R-r)}} $$ |
| </p> | | </p> |
| </p> | | </p> |
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| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$\nu = \frac{\sqrt{v^2+u^2}}{2(R-r)}$$ | | $$\nu = \frac{\sqrt{v^2+u^2}}{2(R-r)}$$ |
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