The total kinetic energy of the rod just after the impulse can be determined by considering the change in linear and rotational kinetic energies.

1. Initial State:
The rod is initially at rest on the horizontal table.

2. Impulse Applied:
A horizontal impulse J is given to the rod at one end. This impulse will generate both linear and angular momentum in the rod.

3. Linear Momentum:
The impulse J will impart linear momentum to the rod. The linear momentum is given by the product of mass (m) and velocity (v). Since the rod is initially at rest, the initial linear momentum is zero.

4. Angular Momentum:
The impulse J will also generate angular momentum in the rod. The angular momentum is given by the product of moment of inertia (I) and angular velocity (ω). The moment of inertia of a rod about its center is (1/12) * m * (2L)^2 = (1/3) * m * L^2. The initial angular velocity is zero.

5. Conservation of Linear and Angular Momentum:
The impulse J will impart linear and angular momentum to the rod. According to the law of conservation of linear and angular momentum, the total initial momentum will be equal to the total final momentum.

6. Final State:
After the impulse, the rod will start moving with a linear velocity v and angular velocity ω.

7. Linear Kinetic Energy:
The final linear kinetic energy can be calculated using the formula (1/2) * m * v^2, where m is the mass of the rod and v is the linear velocity. The linear velocity can be determined by considering the conservation of linear momentum.

8. Angular Kinetic Energy:
The final angular kinetic energy can be calculated using the formula (1/2) * I * ω^2, where I is the moment of inertia of the rod and ω is the angular velocity. The angular velocity can be determined by considering the conservation of angular momentum.

9. Total Kinetic Energy:
The total kinetic energy of the rod just after the impulse is the sum of the linear and angular kinetic energies.

In conclusion, the total kinetic energy of the rod just after the impulse can be determined by considering the linear and angular momentum generated by the impulse, and then using the conservation of momentum to determine the final linear and angular velocities. The final kinetic energy is the sum of the linear and angular kinetic energies.