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| <h3 id="back-link"><a href="../../#1.4">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.4">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.4.15^*.$ There is a bundle of identical nuclei moving with velocity $v$. The nuclei in the beam spontaneously divide into pairs of identical fragments. The velocity of the fragments moving in the direction of the beam is $3v$. Find the velocity of the fragments moving in the direction perpendicular to the beam. | | $1.4.15^*.$ There is a bundle of identical nuclei moving with velocity $v$. The nuclei in the beam spontaneously divide into pairs of identical fragments. The velocity of the fragments moving in the direction of the beam is $3v$. Find the velocity of the fragments moving in the direction perpendicular to the beam. |
| </p> | | </p> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <p>Let's move to the beam center of mass frame.</p> | | <p>Let's move to the beam center of mass frame.</p> |
| <p>In this frame of reference, the relative velocity is related to the velocity in the NFR by the relation</p> | | <p>In this frame of reference, the relative velocity is related to the velocity in the NFR by the relation</p> |
| <p class="exp">$$ \vec{v'} = \vec{v} + \vec{v}_{rel} $$</p> | | <p class="exp">$$ \vec{v'} = \vec{v} + \vec{v}_{rel} $$</p> |
| <p>Where $\vec{v'}$ and $\vec{v}$ are the velocity in the inertial reference frame and the velocity of the reference frame of the system, respectively.</p> | | <p>Where $\vec{v'}$ and $\vec{v}$ are the velocity in the inertial reference frame and the velocity of the reference frame of the system, respectively.</p> |
| <p>By the condition, when $\vec{v}_{rel}$ and $\vec{v}$ are co-directed $v' = 3 v$</p> | | <p>By the condition, when $\vec{v}_{rel}$ and $\vec{v}$ are co-directed $v' = 3 v$</p> |
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| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/1/1.4.15/draw1.png" | | <img src="https://savchenkosolutions.com/1/1.4.15/draw1.png" |
| loading="lazy" width="175" /> | | loading="lazy" width="175" /> |
| <figcaption> | | <figcaption> |
| Representation of $\vec{v'}$ as a sum of two vectors | | Representation of $\vec{v'}$ as a sum of two vectors |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p>From where</p> | | <p>From where</p> |
| <p class="exp">$$v_{rel} = 2v$$</p> | | <p class="exp">$$v_{rel} = 2v$$</p> |
| <p>Next, let's consider the fragments that flew with the speed of $\vec{v}_\perp$)</p> | | <p>Next, let's consider the fragments that flew with the speed of $\vec{v}_\perp$)</p> |
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| <img src="https://savchenkosolutions.com/1/1.4.15/draw.png" | | <img src="https://savchenkosolutions.com/1/1.4.15/draw.png" |
| loading="lazy" width="175" /> | | loading="lazy" width="175" /> |
| <figcaption> | | <figcaption> |
| Vector image | | Vector image |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p>Going back to inertial reference frame, we get that </p> | | <p>Going back to inertial reference frame, we get that </p> |
| <p class="exp">$$\vec{v}_\perp = \vec{v}_{rel}-\vec{v}$$</p> | | <p class="exp">$$\vec{v}_\perp = \vec{v}_{rel}-\vec{v}$$</p> |
| <p>By the Pythagorean theorem</p> | | <p>By the Pythagorean theorem</p> |
| <p class="exp">$$\fbox{${v}_\perp = 3{v}$}$$</p> | | <p class="exp">$$\fbox{${v}_\perp = 3{v}$}$$</p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$\sin \alpha = u/v$$ | | $$\sin \alpha = u/v$$ |
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