| ### Statement | | ### Statement |
| | | |
| $1.4.15^*.$ 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. |
| | | |
| ### Solution | | ### Solution |
| | | |
| Let's move to the beam center of mass frame. | | Let's move to the beam center of mass frame. |
| | | |
| In this frame of reference, the relative velocity is related to the velocity in the NFR by the relation | | In this frame of reference, the relative velocity is related to the velocity in the NFR by the relation |
| | | |
| $$ | | $$ |
| \vec{v'} = \vec{v} + \vec{v}_{rel} | | \vec{v'} = \vec{v} + \vec{v}_{rel} |
| $$ | | $$ |
| | | |
| 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. | | 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. |
| | | |
| By the condition, when $\vec{v}_{rel}$ and $\vec{v}$ are co-directed $v' = 3 v$ | | By the condition, when $\vec{v}_{rel}$ and $\vec{v}$ are co-directed $v' = 3 v$ |
| | | |
|  | |  |
| | | |
| From where | | From where |
| | | |
| $$ | | $$ |
| v_{rel} = 2v | | v_{rel} = 2v |
| $$ | | $$ |
| | | |
| Next, let's consider the fragments that flew with the speed of $\vec{v}_\perp$) | | Next, let's consider the fragments that flew with the speed of $\vec{v}_\perp$) |
| | | |
|  | |  |
| | | |
| Going back to inertial reference frame, we get that | | Going back to inertial reference frame, we get that |
| | | |
| $$ | | $$ |
| \vec{v}_\perp = \vec{v}_{rel}-\vec{v} | | \vec{v}_\perp = \vec{v}_{rel}-\vec{v} |
| $$ | | $$ |
| | | |
