Well, if object A is moving due east and object B is moving due north, then after the collision, they will be stuck together and moving in some new direction. Let's call the magnitude of the total momentum P and the direction θ.
To find P, we can use the fact that momentum is conserved. Since momentum is a vector quantity, we need to consider both the x and y components separately.
In the x direction, the initial momentum is m_A * v_0A, since object A is the only one with momentum in that direction. After the collision, the total momentum in the x direction is (m_A + m_B) * V_x, where V_x is the x-component of the final velocity.
In the y direction, the initial momentum is m_B * v_0B, since object B is the only one with momentum in that direction. After the collision, the total momentum in the y direction is (m_A + m_B) * V_y, where V_y is the y-component of the final velocity.
Since momentum is conserved, the initial momentum in the x direction must be equal to the final momentum in the x direction, and the initial momentum in the y direction must be equal to the final momentum in the y direction.
So, we can set up two equations:
m_A * v_0A = (m_A + m_B) * V_x
m_B * v_0B = (m_A + m_B) * V_y
Now, we can solve these equations to find V_x and V_y, the components of the final velocity.
Once we have V_x and V_y, we can find the magnitude of the total momentum P using the Pythagorean theorem: P = sqrt(V_x^2 + V_y^2).
Finally, we can find the direction θ by using trigonometry. The angle θ is the arctan of V_y/V_x.
Now, I know this may seem like a lot of math, but don't worry, I'll leave you to calculate the final answer. And remember, if you get stuck, just ask for help. Good luck!