A nucleus traveling at speed v splits into two identical fragments. Determine the maximum possible angle \alpha between the velocities of one of the fragments and the vector v, if the fragments have a velocity u < v during the decay of a resting nucleus.

Solutions of Savchenko Problems in Physics

Aliaksandr Melnichenka
October 2023

    <h3 id="back-link"><a href="/#1.4">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>
    <p>
        $1.4.14.$ A nucleus traveling at speed $v$ splits into two identical fragments. Determine the maximum possible angle $\alpha$ between the velocities of one of the fragments and the vector $v$, if the fragments have a velocity $u < v$ during the decay of a resting nucleus.
    </p>

    <h3>Solution</h3>
    <p>
        <p>The velocity and one of the accelerations of the nucleus are</p>

$$\vec{u} = \vec{v}_{0} + \vec{v}$$

where $\vec{v}_{0}$ is its velocity before the decay of the nucleus, $\vec{v}$ is the velocity of the fragment relative to the resting nucleus (that is, in the coordinate system in which the nucleus was at rest before the decay).

Let's arrange the vector $\vec{v}_{0}$ as shown in the figure. The end of the vector can lie on a circle of radius $| \vec{v} |$.

It is clear from the figure that the vector and forms the maximum angle $\alpha$ with the vector $\vec{v}_{0}$ if the angle between the vectors $\vec{u}$ and $\vec{v}$ is $\pi / 2$. In this case

$$\fbox{$ \alpha = \arcsin \frac{v}{v_0} $}$$

    <h4>Answer</h4>
    <p>
        $$\sin \alpha = u/v$$
    </p>


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