In Fig. 9-21, projectile particle 1 is an alpha particle and...

In Fig. $9-21,$ projectile particle 1 is an alpha particle and target particle 2 is an oxygen nucleus.The alpha particle is scattered at angle $\theta_{1}=64.0^{\circ}$ and the oxygen nucleus recoils with speed $1.20 \times$ $10^{5} \mathrm{m} / \mathrm{s}$ and at angle $\theta_{2}=51.0^{\circ} .$ In atomic mass units, the mass of the alpha particle is 4.00 $\mathrm{u}$ and the mass of the oxygen nucleus is 16.0 $\mathrm{u}$ . What are the (a) final and (b) initial speeds of the alpha particle?

In Fig. $9-21,$ projectile particle 1 is an alpha particle and target particle 2 is an oxygen nucleus.The alpha particle is scattered at angle $\theta_{1}=64.0^{\circ}$ and the oxygen nucleus recoils with speed $1.20 \times$ $10^{5} \mathrm{m} / \mathrm{s}$ and at angle $\theta_{2}=51.0^{\circ} .$ In atomic mass units, the mass of the alpha particle is 4.00 $\mathrm{u}$ and the mass of the oxygen nucleus is 16.0 $\mathrm{u}$ .
What are the (a) final and (b) initial speeds of the alpha particle?

Key Concepts

Conservation of Momentum

In collision problems, the total momentum of the system is conserved in all directions. This means that the vector sum of the momenta of all particles before the collision equals the vector sum after the collision. Applying this principle separately in each coordinate direction (usually the horizontal and vertical components) provides a set of equations that are essential for determining unknown velocities or directions after the collision.

Conservation of Kinetic Energy

For elastic collisions, the total kinetic energy of the system is conserved along with momentum. This conservation principle yields an additional equation relating the speeds of the colliding particles before and after the collision. When combined with the momentum conservation equations, it allows for solving for unknown speeds in the system.

Scattering Angles in Collisions

The scattering angles, which indicate the deviation of particle trajectories after a collision from the initial direction, are key to resolving the momentum vectors into directional components. Understanding these angles is critical for correctly applying momentum conservation in each coordinate direction and for accurately determining the relationships between the various speeds and directions involved.

Elastic Collision Dynamics

Elastic collisions are characterized by both momentum and kinetic energy being conserved. The dynamics of such collisions are analyzed using the conservation principles along with the geometric arrangement provided by the scattering angles. This analysis is fundamental in determining how the initial motion is redistributed among the colliding particles, particularly in two-dimensional collision processes.

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