1. **Problem Statement:** Determine the speed of block B when the end of the cord at A is pulled down with a speed of 2 m/s. 2. **Understanding the system:** The cord passes over pulleys D, C, and E, with block B hanging from pulley E. Pulling the cord at A changes the lengths of cord segments, affecting the speed of block B. 3. **Key principle:** The total length of the cord is constant. Let $s_A$, $s_B$, and $s_C$ be the lengths of the cord segments as labeled. 4. **Relationship between lengths:** Assuming the cord passes over pulleys such that the length changes satisfy: $$s_A + 2s_B + s_C = \text{constant}$$ 5. **Differentiating with respect to time $t$:** $$\frac{d}{dt}(s_A) + 2\frac{d}{dt}(s_B) + \frac{d}{dt}(s_C) = 0$$ 6. **Given:** The speed at A is $v_A = \frac{ds_A}{dt} = -2$ m/s (negative because pulled down, assuming upward positive). 7. **Assuming $s_C$ is constant or its rate is zero:** $$\frac{ds_C}{dt} = 0$$ 8. **Substitute into the differentiated equation:** $$-2 + 2\frac{ds_B}{dt} + 0 = 0$$ 9. **Solve for $\frac{ds_B}{dt}$:** $$2\frac{ds_B}{dt} = 2$$ $$\frac{ds_B}{dt} = 1$$ 10. **Interpretation:** The speed of block B, $v_B = \frac{ds_B}{dt} = 1$ m/s upward. **Final answer:** $$\boxed{v_B = 1 \text{ m/s upward}}$$