(a) Using the conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.

Before the collision, the momentum is:

\(km \times 6 - m \times 2\)

After the collision, the momentum is:

\((km + m) \times 4\)

Equating the two expressions:

\(km \times 6 - m \times 2 = (km + m) \times 4\)

\(6km - 2m = 4km + 4m\)

\(2km = 6m\)

\(k = 3\)

(b) The initial kinetic energy (KE) is given by:

\(\frac{1}{2} \times km \times 6^2 + \frac{1}{2} \times m \times (-2)^2\)

\(= 18km + 2m\)

The kinetic energy after the collision is:

\(\frac{1}{2} \times (km + m) \times 4^2\)

\(= 8(km + m)\)

\(= 8km + 8m\)

The loss of kinetic energy is:

\((18km + 2m) - (8km + 8m)\)

\(= 10km - 6m\)

Substituting \(k = 3\):

\(= 30m - 6m\)

\(= 24m\)