(a) Begin by drawing a picture of the situation with the initial momentum of each puck and the final momentum of the system.
The momentum of pucks A and B before and after the collision.
To sketch the final momentum, you must actually apply the Momentum Principle to the system. First, define the system as pucks A and B (together). The Momentum Principle states
Because the net external force on the system is zero during the collision (after all, they are moving on a low friction air hockey table that is level), the change in the momentum of the system is zero, and the momentum of the system is constant as shown below.
In the picture the initial momentum of the system is the sum of the initial momentum vector for each puck. Sketch the initial momentum of each puck, tail to head. The resultant (sum) is the vector from the tail of the first vector to the head of the second vector, as shown below. Note that the final momentum of the system is equal to the initial momentum of the system.
The initial and final momentum of the system are equal.
Now, solve it algebraically. The total momentum of the system is the sum of the momentum of each puck in the system. Use the nonrelativistic expression for momentum in terms of velocity.
Because the pucks stick together, they have the same final velocity, so you can write and just call it and factor it out of the two terms on the left side of the above equation.
Substitute given quantities to get the final velocity.
Note that the final momentum of the system is to the right and downward, exactly as shown in the picture.