Assertion The total kinetic energy of a rolling solid sphere is the sum of translational and rotationla kinetic energies .
Reason For all solid bodies. Totla kinetic energy is always twice of translational kinetic energy.

Options

A. Both assertion and reson are true and reason is the correct explanation of assertion
B. Both assetion and reason are true but reason is not the correct explanation of assertion
C. Assertion is true but reason is false
D. Both assetion and reason are flase

Answer

To solve the given assertion and reason question, we will analyze both the assertion and the reason step by step.

### Step 1: Understanding the Assertion
The assertion states that "the total kinetic energy of a rolling solid sphere is the sum of translational and rotational kinetic energies."

- **Translational Kinetic Energy (TKE)** is given by the formula:
\[ TKE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the sphere and \( v \) is its linear velocity.

- **Rotational Kinetic Energy (RKE)** is given by the formula:
\[ RKE = \frac{1}{2} I \omega^2 \]
where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.

For a solid sphere, the moment of inertia \( I \) is:
\[ I = \frac{2}{5} m r^2 \]
where \( r \) is the radius of the sphere. The angular velocity \( \omega \) can be related to the linear velocity \( v \) by:
\[ \omega = \frac{v}{r} \]

### Step 2: Summing Kinetic Energies
Now, we can express the total kinetic energy (KE) of the rolling sphere:
\[ KE = TKE + RKE \]
Substituting the formulas we have:
\[ KE = \frac{1}{2} mv^2 + \frac{1}{2} \left(\frac{2}{5} m r^2\right) \left(\frac{v}{r}\right)^2 \]
This simplifies to:
\[ KE = \frac{1}{2} mv^2 + \frac{1}{2} \left(\frac{2}{5} m r^2\right) \left(\frac{v^2}{r^2}\right) \]
\[ = \frac{1}{2} mv^2 + \frac{1}{5} mv^2 \]
\[ = \frac{5}{10} mv^2 + \frac{2}{10} mv^2 = \frac{7}{10} mv^2 \]

Thus, the assertion is true: the total kinetic energy of a rolling solid sphere is indeed the sum of translational and rotational kinetic energies.

### Step 3: Understanding the Reason
The reason states that "for all solid bodies, total kinetic energy is always twice the translational kinetic energy."

This statement is incorrect because the relationship between total kinetic energy and translational kinetic energy depends on the moment of inertia and the radius of gyration of the specific solid body. The total kinetic energy can vary based on the shape and mass distribution of the body. For example, for a solid sphere, the total kinetic energy is not necessarily twice the translational kinetic energy.

### Conclusion
- **Assertion**: True
- **Reason**: False

Thus, the correct answer is that the assertion is true, but the reason is false.