To solve this problem we will apply the concepts related to the conservation of momentum. Momentum can be defined as the product between mass and velocity. We will depart to facilitate the understanding of the demonstration, considering the initial and final momentum separately, but for conservation, they will be later matched. Thus we will obtain the value of the mass. Our values will be defined as
[tex]m_1 = m[/tex]
[tex]m_2 = M[/tex]
[tex]v_{1i} =v_0[/tex]
[tex]v_{2i} = 0[/tex]
Initial momentum will be
[tex]P_i = m_iv_{1i}+m_2v_{2i}[/tex]
[tex]P_i = mv_0[/tex]
After collision
[tex]v_{1f} = v_{2f} = \frac{v_0}{3}[/tex]
Final momentum
[tex]P_f = (m_1+m_2)(\frac{v_0}{3})[/tex]
[tex]P_f = (m+M)(\frac{v_0}{3})[/tex]
From conservation of momentum
[tex]P_f = P_i[/tex]
Replacing,
[tex](m+M)(\frac{v_0}{3})=mv_0[/tex]
[tex](m+M)\frac{1}{3} = m[/tex]
[tex]m+M=3m[/tex]
[tex]M=3m-m[/tex]
[tex]M=2m[/tex]
