Answer
The acceleration of the 2.0-kg block is $3.27~m/s^2$.
Work Step by Step
We can set up a force equation for the 1.0-kg block. Let $m_2$ be the mass of this block. Let $T$ be the tension in the rope;
$\sum F = m_2~a_2$
$m_2~g - T = m_2~a_2$
$T = m_2~g - m_2~a_2$
We can use this expression for the tension $T$ in the force equation for 2.0-kg block. Let $m_1$ be the mass of this block. Note that $a_2 = 2a_1$.
$\sum F = m_1~a_1$
$2T = m_1~a_1$
$2m_2~g - 2m_2~a_2 = m_1~a_1$
$2m_2~g - 2m_2~(2~a_1) = m_1~a_1$
$2m_2~g = m_1~a_1+4m_2~a_1$
$a_1 = \frac{2m_2~g}{m_1+4m_2}$
$a_1 = \frac{(2)(1.0~kg)(9.80~m/s^2)}{2.0~kg+(4)(1.0~kg)}$
$a_1 = 3.27~m/s^2$
The acceleration of the 2.0-kg block is $3.27~m/s^2$.
