Answer
$8.62m/s$
Work Step by Step
The required velocity can be determined as follows:
According to the law of conservation of energy
$\frac{1}{2}(m_t+m_g+m_b) v_1^2+(m_t+m_g+m_b) g_h=\frac{1}{2}(m_t+m_g+m_b) v_2^2+0$
This simplifies to:
$v_2=7.672m/s$
Now we use the principle of impulse and momentum in the x-direction
$mv_{x_1}+\Sigma \int F_xdt=mv_{x_2}$
$\implies (m_t+m_g+m_b) v_2=(m_t+m_g)v_t+m_bv_b$
We plug in the known values to obtain:
$(10+40+45)(7.672)=(10+40)v_t+45v_b$
but $v_b=v_t-2$
$\implies (10+40+45)(7.672)=(10+40)v_t+45(v_t-2)$
This simplifies to:
$v_t=8.62m/s$
