Showing that KE is minimized when objects stick together in a collision

[Azu_]

Homework Statement

This problem deals with an inelastic collision, where Mass A is given an initial velocity and collides with Mass B which is initially at rest.

The equation I'm given is

K_f = \frac{1}{2} m_A (V_Ax^2 + V_Ay^2) + \frac{1}{2} m_B (V_Bx^2 + V_By^2)
The x's and y's are part of the subscript.

So the problem says:

Start with the expression shown above for the system's kinetic energy after the collision. Energy is not a vector, but the final total speeds are expressed with x and y components, by Pythagorean theorem. This notation puts them on a comparable footing with momentum, which is a vector.

Show that this quantity is minimized when the objects stick together.

The following steps will walk you through it:

a) Take the derivative with respect to anyone of the four velocity variables (your choice).

b) The x and y directions are independent, i.e. if you pick an x variable, the derivatives of the y variables are 0.

c) The two variables along the same axis are NOT independent. Use momentum to make a substitution. This should lead to some clear conclusions.

d) Now take the 2nd derivative to get the concavity. What sign should it be?

e) Without recomputing, extrapolate what you would have gotten if you had chosen a variable from the other axis in step a (y instead of x, etc.). Why is this step necessary to demonstrate that the objects truly do "stick together"? The attempt at a solution

So for part a, I chose to take the derivative with respect to Vax and I get
dk/dVa = m_A V_A + m_B V_B (dV_B/dV_A)

Now I'm really lost on part c. I'm really unclear as to how to use momentum and what to substitute in.

 

[haruspex]

I'm really unclear as to how to use momentum and what to substitute in.

Momentum is a conserved vector, so it is conserved in each of the directions, x and y, separately. Write down two equations expressing this.