1. **State the problem:** We have a cart of mass $m_1 = 0.8$ kg on a frictionless table connected by a string over a pulley to a hanging mass $m_2 = 0.2$ kg. We want to find the magnitude of the acceleration $a$ of both masses. 2. **Set up the forces and equations:** - For the cart ($m_1$), the only horizontal force is the tension $T$ in the string, so Newton's second law gives: $$m_1 a = T$$ - For the hanging mass ($m_2$), the forces are gravity downward and tension upward: $$m_2 g - T = m_2 a$$ where $g = 9.8$ m/s$^2$ is acceleration due to gravity. 3. **Combine the equations:** From the first equation, $T = m_1 a$. Substitute into the second: $$m_2 g - m_1 a = m_2 a$$ Rearranged: $$m_2 g = m_1 a + m_2 a = a(m_1 + m_2)$$ 4. **Solve for acceleration $a$:** $$a = \frac{m_2 g}{m_1 + m_2}$$ Substitute values: $$a = \frac{0.2 \times 9.8}{0.8 + 0.2} = \frac{1.96}{1.0} = 1.96$$ 5. **Interpretation:** The acceleration of both the cart and the hanging mass is $1.96$ m/s$^2$. This means both move with the same magnitude of acceleration due to the string connecting them. **Final answer:** $$\boxed{a = 1.96 \text{ m/s}^2}$$ This process uses Newton's second law and the fact that the string tension is the same on both sides of the pulley, assuming no friction or mass in the pulley or string.