Question

Classically, what happens when a moving object has a head-on elastic collision with a stationary object of exactly equal mass? What if it strikes an object of smaller mass? Of larger mass? How do these ideas relate to Rutherford’s conclusion about the nature of the atom?

Short answer

Rutherford suggested that the positive charge and mass of the particle are moved in a small volume in the molecule. This small enormous article was subsequently named "the core".

Step-by-step solution

Define conservation of energy and momentum

In physical science and science, the law of preservation of energy expresses that the all-out energy of a segregated framework stays steady; being rationed over the long haul is said.

The law of protection of energy expresses that in a detached framework, the absolute force of at least two bodies following up on one another remaining parts is consistent except if an outside force is applied. Subsequently, energy cannot be made nor annihilated.

Calculation of the conversation of energy and momentum

Let the diagram of the before and after the elastic collision between two objects, with the target initially at rest, be,

Apply the conversation of energy and momentum for an elastic collision between a moving object and a stationary object.

Conservation of momentum:

$m_p{v}_1+m_t{v}_2=m_p{v}_1^{\text{'}}+m_t{v}_2^{\text{'}}\left(\because v_2=0\right)$

$m_p{v}_1+m_t{v}_1^{\text{'}}=m_t{v}_2^{\text{'}}$ … (1)

Conservation of energy:

$\frac{1}{2}{m}_p{V}_1^2=\frac{1}{2}{m}_p{v}_1^{\text{'}2}+\frac{1}{2}{m}_t{V^{\text{'}2}}_2$ … (2)

By using the equation (1) and (2),

$$\begin{gathered} v_1^{\text{'}}=\left(\frac{m_p-m_t}{m_p+m_t}\right)v_1 \\ {v}_2^{\text{'}}=\left(\frac{2m_p}{m_p+m_t}\right)v_1 \end{gathered}$$ … (3)

The three cases for a head-on elastic collision

There are three cases of a head-on elastic collision between a projectile and a stationary target.

Case 1: A projectile and a stationary target have the same mass$\left(m_p=m_t=m\right)$.

$$\begin{gathered} v_1^{\text{'}}=\left(\frac{m-m}{m+m}\right)v_1 \\ {v}_2^{\text{'}}=\left(\frac{2m}{m+m}\right)v_1 \end{gathered}$$

$\therefore \begin{array}{c}{v}_1^{\text{'}}=0\\ v_2^{\text{'}}=v_1\end{array}$

Hence, the projectile comes to rest, and the stationary target starts moving at the same speed as the initial speed of the projectile.

Case 2: A stationary target is much smaller than a projectile $(m_p\gg m_t)$.

$$\begin{gathered} v_1^{\text{'}}\simeq \left(\frac{m_p}{m_p}\right)v_1 \\ {v}_2^{\text{'}}\simeq \left(\frac{2m_p}{m_p}\right)v_1 \end{gathered}$$

$\therefore \begin{array}{c}{v}_1^{\text{'}}\simeq v_1\\ v_2^{\text{'}}\simeq 2v_1\end{array}$

Hence, the projectile remains unchanged, and the stationary target starts moving almost twice the initial speed of the projectile.

Case 3: A stationary target is much larger than a projectile $(m_p\ll m_t)$.

$$\begin{gathered} v_1^{\text{'}}\simeq \left(\frac{-m_t}{m_t}\right)v_1 \\ {v}_2^{\text{'}}\simeq \left(\frac{2\left(0\right)}{m_t}\right)v_1 \end{gathered}$$

$\therefore \begin{array}{c}{v}_1^{\text{'}}\simeq v_1\\ v_2^{\text{'}}\simeq 0\end{array}$

Hence, the projectile will bounce back with almost the same speed, and the target remains stationary.

Rutherford’s conclusion about the nature of the atom

Rutherford's examination included besieging a dainty gold foil with alpha particles (which are emphatically charged). One of the perceptions was that a small number of alpha particles bobbed in reverse (in reverse dissipating).

This dispersing is because of Coulomb communications, aversion between the alpha particles, and the positive charges in the core. As needs are, no energy is lost as intensity or sound energies since there is no real contact, and this can be considered a versatile impact between a moving item and a fixed huge item (case (3)).

The little number of back dissipated alpha particles shows that the objective must be tiny in size as the likelihood of being on the impact course is minuscule.

In the end, Rutherford suggested that the positive charge and mass of the particle are moved in a small volume in the molecule. This small enormous article was subsequently named "the core".