A radioactive nucleus of mass `M` emits a photon of frequency `v` and the nucleus recoils. The recoil energy will be

Options

A. `h^2 V^2//2 Mc^2`
B. zero
C. hv
D. `Mc^2 - hv`

Explanation

To find the recoil energy of a radioactive nucleus that emits a photon of frequency \( v \), we can follow these steps:

### Step 1: Understand the Conservation of Momentum
When the nucleus emits a photon, the total momentum before and after the emission must be conserved. Initially, both the nucleus and the photon are at rest, so the total initial momentum is zero.

### Step 2: Define the Momentum of the Photon
The momentum \( P \) of a photon can be expressed using the formula:
\[ P = \frac{E}{c} \]
where \( E \) is the energy of the photon and \( c \) is the speed of light. The energy of the photon can be expressed in terms of its frequency \( v \) as:
\[ E = h v \]
where \( h \) is Planck's constant. Therefore, the momentum of the photon becomes:
\[ P_{\text{photon}} = \frac{h v}{c} \]

### Step 3: Apply Conservation of Momentum
Let \( P_{\text{nucleus}} \) be the momentum of the recoiling nucleus. According to the conservation of momentum:
\[ P_{\text{nucleus}} + P_{\text{photon}} = 0 \]
This implies:
\[ P_{\text{nucleus}} = -P_{\text{photon}} = -\frac{h v}{c} \]

### Step 4: Relate Momentum to Recoil Energy
The momentum of the recoiling nucleus can also be expressed in terms of its mass \( M \) and its velocity \( v_r \):
\[ P_{\text{nucleus}} = M v_r \]
Setting the two expressions for momentum equal gives:
\[ M v_r = \frac{h v}{c} \]
From this, we can solve for the recoil velocity \( v_r \):
\[ v_r = \frac{h v}{M c} \]

### Step 5: Calculate the Recoil Energy
The kinetic energy \( E_r \) of the recoiling nucleus can be expressed as:
\[ E_r = \frac{1}{2} M v_r^2 \]
Substituting the expression for \( v_r \):
\[ E_r = \frac{1}{2} M \left(\frac{h v}{M c}\right)^2 \]
This simplifies to:
\[ E_r = \frac{1}{2} M \cdot \frac{h^2 v^2}{M^2 c^2} = \frac{h^2 v^2}{2M c^2} \]

### Final Result
Thus, the recoil energy of the nucleus when it emits a photon of frequency \( v \) is:
\[ E_r = \frac{h^2 v^2}{2M c^2} \]