Two spheres P and Q of equal mass m collide. The initial velocities are uP = au(-4i + 2j) and uQ = au(2i + j). The line of centres is parallel to i. Find the final velocities.

Step 1: Determine the components of the initial velocities.
The initial velocities of spheres P and Q are given as:

$$\mathbf{u}_P = \mu(-4\mathbf{i} + 2\mathbf{j}) = -4\mu\mathbf{i} + 2\mu\mathbf{j}$$ $$\mathbf{u}_Q = \mu(2\mathbf{i} + \mathbf{j}) = 2\mu\mathbf{i} + \mu\mathbf{j}$$

The line of centres during the collision is parallel to the vector $\mathbf{i}$. This means that the components of velocity perpendicular to the line of centres (the $\mathbf{j}$ components) remain unchanged. The collision only affects the components of velocity parallel to the line of centres (the $\mathbf{i}$ components).

Let the final velocities be $\mathbf{v}_P = v_{Px}\mathbf{i} + v_{Py}\mathbf{j}$ and $\mathbf{v}_Q = v_{Qx}\mathbf{i} + v_{Qy}\mathbf{j}$.
From the unchanged perpendicular components:

$$v_{Py} = u_{Py} = 2\mu$$ $$v_{Qy} = u_{Qy} = \mu$$

Step 2: Apply the conservation of momentum and the coefficient of restitution along the line of centres (x-direction).
The masses of both spheres are $m$.
Conservation of momentum along the x-axis:

$$m u_{Px} + m u_{Qx} = m v_{Px} + m v_{Qx}$$ $$$$