1. Meaning of Q-value

For a nuclear reaction of the form $a + A \rightarrow b + B,$ the Q-value is defined as $Q = (m_{\text{initial}} - m_{\text{final}})c^2.$

It represents the net energy released in the reaction.

• If $Q>0$, the reaction is exoergic.
• If $Q<0$, the reaction is endoergic.

2. Energy conservation in a nuclear reaction

If the target nucleus is initially at rest, then the total initial kinetic energy is just the kinetic energy of the projectile: $K_p.$

After the reaction, the total final kinetic energy becomes $K_{\text{final}} = K_p + Q.$

This follows from conservation of energy: $\text{initial kinetic energy} + \text{released energy} = \text{final kinetic energy}.$

So, $K_{\text{final}} = K_p + Q.$

3. Physical condition on kinetic energy

Kinetic energy cannot be negative. Therefore, $K_{\text{final}} > 0$ for the reaction products to exist physically.

Hence, $K_p + Q > 0.$

4. Check the options

• A: $Q = K_p$
  Not a general relation. False.

• B: $(K_p + Q) < 0$
  Would imply negative final kinetic energy, impossible. False.

• C: The printed option appears incomplete/incorrectly rendered, so it is not a valid general relation.

• D: $(K_p + Q) > 0$
  Correct physical condition.

5. Final answer

The correct relation is $K_p + Q > 0.$