To solve this problem, we will apply the principle of conservation of momentum. Since the spheres are colliding on a frictionless surface, we know that the total momentum before the collision must equal the total momentum after the collision.

Step 1: Define the known variables

• Mass of sphere A, $m_A = 0.020 \, \text{kg}$
• Mass of sphere B, $m_B = 0.030 \, \text{kg}$
• Mass of sphere C, $m_C = 0.050 \, \text{kg}$
• The final velocity of all three spheres combined is $v_{\text{final}} = 0.50 \, \text{m/s}$ in the +x-direction.

The initial velocities of A and B are given in the problem, but we need to deduce the x-component of C's initial velocity. Let’s assume:

• The initial velocity of A in the x-direction is $v_{A_x}$ (given).
• The initial velocity of B in the x-direction is $v_{B_x}$ (given).
• The initial velocity of C in the x-direction is $v_{C_x}$, which we need to find.

Step 2: Apply the conservation of momentum in the x-direction

The momentum before the collision is the sum of the momenta of the three spheres:

$\text{Total initial momentum in the x-direction} = m_A v_{A_x} + m_B v_{B_x} + m_C v_{C_x}$

After the collision, all three spheres move together with velocity $v_{\text{final}}$. The total momentum in the x-direction after the collision is:

$\text{Total final momentum in the x-direction} = (m_A + m_B + m_C) v_{\text{final}}$

Since momentum is conserved, we equate the initial and final momenta:

$m_A v_{A_x} + m_B v_{B_x} + m_C v_{C_x} = (m_A + m_B + m_C) v_{\text{final}}$

Step 3: Solve for $v_{C_x}$

Rearranging the equation to solve for $v_{C_x}$:

$v_{C_x} = \frac{(m_A + m_B + m_C) v_{\text{final}} - m_A v_{A_x} - m_B v_{B_x}}{m_C}$

Step 4: Plug in the known values

Now, substitute the known values for $m_A$, $m_B$, $m_C$, $v_{\text{final}}$, and the initial velocities of A and B:

• $m_A = 0.020 \, \text{kg}$
• $m_B = 0.030 \, \text{kg}$
• $m_C = 0.050 \, \text{kg}$
• $v_{\text{final}} = 0.50 \, \text{m/s}$
• $v_{A_x}$ and $v_{B_x}$ are the initial velocities of A and B in the x-direction (which should be given in the problem or need to be inferred).

Once you plug in all the values, you can calculate the required $v_{C_x}$.

Do you have the initial velocities of spheres A and B? If you provide them, I can give you the exact value for $v_{C_x}$.