`_83^212Bi` decays as per following equation.
`_83^212Birarr_82^208Ti+_2^4He`
The kinetic energy of `alpha`-particle emitted is 6.802 MeV. Calculate the kinetic energy of Ti recoil atoms.

Explanation

To solve the problem, we will use the conservation of momentum and the relationship between kinetic energy and momentum. Here are the steps to calculate the kinetic energy of the Ti recoil atoms:

### Step-by-Step Solution:

1. **Identify the decay reaction**:
The decay of bismuth-212 (_83^212Bi) can be represented as:
\[ _83^{212}Bi \rightarrow _82^{208}Ti + _2^4He \]
Here, the bismuth nucleus decays into a titanium nucleus and an alpha particle (helium nucleus).

2. **Understand the conservation of momentum**:
Since the bismuth nucleus is initially at rest, the total initial momentum is zero. According to the conservation of momentum:
\[ p_{initial} = p_{final} \]
This means that the momentum of the titanium nucleus (Ti) and the alpha particle (He) must balance each other out.

3. **Set up the momentum equation**:
Let \( m_{Ti} \) be the mass of the titanium nucleus, \( m_{\alpha} \) be the mass of the alpha particle, \( KE_{Ti} \) be the kinetic energy of the titanium nucleus, and \( KE_{\alpha} \) be the kinetic energy of the alpha particle. The momentum can be expressed as:
\[ m_{Ti} \cdot v_{Ti} = m_{\alpha} \cdot v_{\alpha} \]
where \( v_{Ti} \) and \( v_{\alpha} \) are the velocities of the titanium and alpha particles, respectively.

4. **Relate kinetic energy to momentum**:
The kinetic energy (KE) is related to momentum (p) by the equation:
\[ KE = \frac{p^2}{2m} \]
Therefore, we can express the momentum in terms of kinetic energy:
\[ p = \sqrt{2m \cdot KE} \]

5. **Substitute the momentum expressions**:
Substitute the expressions for momentum into the conservation of momentum equation:
\[ m_{Ti} \cdot \sqrt{2m_{Ti} \cdot KE_{Ti}} = m_{\alpha} \cdot \sqrt{2m_{\alpha} \cdot KE_{\alpha}} \]

6. **Simplify the equation**:
Squaring both sides and simplifying gives:
\[ m_{Ti}^2 \cdot 2KE_{Ti} = m_{\alpha}^2 \cdot 2KE_{\alpha} \]
This simplifies to:
\[ KE_{Ti} = \frac{m_{\alpha}}{m_{Ti}} \cdot KE_{\alpha} \]

7. **Insert known values**:
We know:
- Mass of alpha particle, \( m_{\alpha} = 4 \) (in atomic mass units)
- Mass of titanium, \( m_{Ti} = 208 \) (in atomic mass units)
- Kinetic energy of alpha particle, \( KE_{\alpha} = 6.802 \) MeV

Plugging these values into the equation:
\[ KE_{Ti} = \frac{4}{208} \cdot 6.802 \]

8. **Calculate the kinetic energy of Ti**:
\[ KE_{Ti} = \frac{4 \times 6.802}{208} = \frac{27.208}{208} \approx 0.1308 \text{ MeV} \]

### Final Answer:
The kinetic energy of the titanium recoil atoms is approximately **0.1308 MeV**.