Introduction:
When a bullet passes through a plank that is free to move, both the bullet and the plank experience a change in velocity. The bullet's initial velocity is u and its final velocity is fu, while the plank's initial velocity is zero. We need to determine the velocity of the bullet relative to the plank after passing through the plank.

Explanation:
To solve this problem, we can apply the principle of conservation of momentum. According to this principle, the total momentum before the interaction is equal to the total momentum after the interaction, provided there are no external forces acting on the system.

Step 1: Conservation of momentum:
The initial momentum of the system is given by the sum of the momenta of the bullet and the plank:
Initial momentum = mass of bullet * initial velocity of bullet + mass of plank * initial velocity of plank

Since the initial velocity of the plank is zero, the initial momentum simplifies to:
Initial momentum = mass of bullet * initial velocity of bullet

Step 2: Final momentum:
The final momentum of the system is given by the sum of the momenta of the bullet and the plank after passing through the plank:
Final momentum = mass of bullet * final velocity of bullet + mass of plank * final velocity of plank

Since the mass of the plank is equal to the mass of the bullet, the final momentum simplifies to:
Final momentum = mass of bullet * final velocity of bullet + mass of bullet * final velocity of plank

Step 3: Equating initial and final momentum:
According to the principle of conservation of momentum, the initial momentum and final momentum should be equal. Therefore, we can equate the two expressions obtained in step 1 and step 2:
mass of bullet * initial velocity of bullet = mass of bullet * final velocity of bullet + mass of bullet * final velocity of plank

Step 4: Solving for final velocity of bullet:
Rearranging the equation, we can solve for the final velocity of the bullet:
mass of bullet * initial velocity of bullet - mass of bullet * final velocity of bullet = mass of bullet * final velocity of plank

mass of bullet * (initial velocity of bullet - final velocity of bullet) = mass of bullet * final velocity of plank

Cancelling the mass of the bullet from both sides of the equation, we get:
initial velocity of bullet - final velocity of bullet = final velocity of plank

Step 5: Final result:
Finally, we can rearrange the equation to isolate the final velocity of the bullet:
final velocity of bullet = initial velocity of bullet - final velocity of plank

Therefore, the velocity of the bullet relative to the plank after passing through the plank is given by the difference between the initial velocity of the bullet and the final velocity of the plank.

Conclusion:
The velocity of the bullet relative to the plank can be determined by subtracting the final velocity of the plank from the initial velocity of the bullet. By applying the principle of conservation of momentum, we can analyze the interaction between the bullet and the plank and determine the final velocities of both objects.