To solve this problem we will apply the concepts related to the conservation of momentum. The momentum can be defined as the product between the mass of the object and its velocity, and the conservation of the momentum as the equality between the change of the initial momentum versus the final momentum. Mathematically, this relationship can be described as
[tex]m_1u_1+m_2u_2 = m_1v_2+m_2v_2[/tex]
Here,
[tex]m_{1,2}[/tex] = Mass of each object
[tex]u_{1,2}[/tex] = Initial velocity of each object
[tex]v_{1,2}[/tex] = Final velocity of each object
According to the statement one of the bodies does not have initial velocity, therefore said term would be zero. And the equation could be rewritten as,
[tex]m_1u_1= m_1v_2+m_2v_2[/tex]
Replacing the values respectively (The mass of your body with its respective speed we would have)
[tex]2kg(u_1) = 2kg(0.3m/s)+1kg(0.5m/s)[/tex]
[tex]u_1 = \frac{2kg(0.3m/s)+1kg(0.5m/s)}{2kg}[/tex]
[tex]u_1 = 0.55m/s[/tex]
Therefore the initial velocity of the 2kg cart is 0.55m/s
