Two identical spheres `A` and `B` are free to move and I, rotate about their centres. They are given the same impel `J`. The lines of action of the impulses pass through tht centre of `A` and away from the centre of `B`, then

Options

A. `A` and `B` will have the same speed
B. `B` will have greater kinetic energy than `A`
C. they will have the same kinetic energy, but the Linear kinetic energy of `B` will be less than that of `A`
D. the kinetic energy of `B` will depend on the point impact of the impulse on `B`

Explanation

To solve the problem, we need to analyze the situation involving two identical spheres, A and B, which are subjected to the same impulse J. The lines of action of the impulses pass through the center of sphere A and away from the center of sphere B. We will evaluate the statements provided in the question step by step.

### Step 1: Understanding the Impulse and its Effects
Impulse (J) is defined as the change in momentum of an object. For both spheres, since they are identical and initially at rest, the impulse will cause them to gain the same linear momentum.

- **For Sphere A**: The impulse acts through its center, resulting in a linear velocity \( v_A \).
- **For Sphere B**: The impulse acts away from its center, which means it will not only gain linear momentum but also experience a torque that will cause it to rotate.

### Step 2: Analyzing Linear Velocities
Since both spheres experience the same impulse J, we can express the change in momentum as:
\[ J = m v \]
where \( m \) is the mass of the spheres and \( v \) is their velocity.

Since both spheres are identical:
- Sphere A will have a linear velocity \( v_A = \frac{J}{m} \).
- Sphere B will also have a linear velocity \( v_B \), but we need to account for the torque and rotation.

### Step 3: Kinetic Energy Comparison
The kinetic energy (KE) of each sphere can be expressed as follows:
- **For Sphere A** (only translational motion):
\[ KE_A = \frac{1}{2} m v_A^2 \]

- **For Sphere B** (translational and rotational motion):
\[ KE_B = \frac{1}{2} m v_B^2 + \frac{1}{2} I \omega^2 \]
where \( I \) is the moment of inertia of sphere B and \( \omega \) is its angular velocity.

### Step 4: Evaluating Statements
1. **Statement A**: "A and B will have the same speed."
- **True**: Both spheres experience the same impulse, leading to the same linear speed for sphere A. Sphere B's speed will be affected by its rotation, but the linear component due to the impulse will be the same.

2. **Statement B**: "B will have greater kinetic energy than A."
- **True**: Sphere B has both translational and rotational kinetic energy, while sphere A has only translational kinetic energy. Therefore, \( KE_B > KE_A \).

3. **Statement C**: "They will have the same kinetic energy."
- **False**: As established, sphere B has additional rotational kinetic energy, so they cannot have the same kinetic energy.

4. **Statement D**: "The kinetic energy of B will depend on the point of impact of the impulse on B."
- **True**: The point of application of the impulse affects the torque and thus the angular velocity \( \omega \), which influences the rotational kinetic energy of sphere B.

### Conclusion
From the analysis, we conclude that:
- Statements A, B, and D are true.
- Statement C is false.