Step-by-step Solution

1. Identify Given Information and Sign Conventions:

   • The mirror is a convex mirror.
   • Radius of curvature, R = 20 m. For a convex mirror, the radius of curvature is positive, so R = +20 m.
   • Initial image position, $v_1 = {{25} \over 3}$ m. For a convex mirror, the image formed is virtual and behind the mirror. According to the Cartesian sign convention, this distance is positive. So, $v_1 = +{{25} \over 3}$ m.
   • Final image position, $v_2 = {{50} \over 7}$ m. Similarly, $v_2 = +{{50} \over 7}$ m.
   • Time interval, $\Delta t = 30$ s.
   • The object is real, so its position u will be negative.

2. Calculate the Focal Length (f): The focal length of a spherical mirror is half its radius of curvature. $f = {R \over 2}$ $f = {+20 \over 2} = +10 \text{ m}$

3. Determine the Initial Object Position (u1): We use the mirror formula: ${{1} \over v} + {{1} \over u} = {{1} \over f}$. For the initial position: ${{1} \over v_1} + {{1} \over u_1} = {{1} \over f}$ ${{1} \over {25/3}} + {{1} \over u_1} = {{1} \over 10}$ ${{3} \over {25}} + {{1} \over u_1} = {{1} \over 10}$ ${{1} \over u_1} = {{1} \over 10} - {{3} \over {25}}$ To subtract the fractions, we find a common denominator, which is 50. ${{1} \over u_1} = {{5} \over 50} - {{6} \over 50} = -{{1} \over 50}$ $u_1 = -50 \text{ m}$ The negative sign indicates that the object is in front of the mirror, as expected for a real object.

4. Determine the Final Object Position (u2): Using the mirror formula for the final position: ${{1} \over v_2} + {{1} \over u_2} = {{1} \over f}$ ${{1} \over {50/7}} + {{1} \over u_2} = {{1} \over 10}$ ${{7} \over {50}} + {{1} \over u_2} = {{1} \over 10}$ ${{1} \over u_2} = {{1} \over 10} - {{7} \over {50}}$ The common denominator is 50. ${{1} \over u_2} = {{5} \over 50} - {{7} \over 50} = -{{2} \over 50} = -{{1} \over 25}$ $u_2 = -25 \text{ m}$

5. Calculate the Distance Traveled by the Object: The object moves from $u_1 = -50$ m to $u_2 = -25$ m. The distance traveled is the magnitude of the displacement. $d = |u_2 - u_1| = |-25 - (-50)| = |-25 + 50| = 25 \text{ m}$

6. Calculate the Speed of the Object: The speed is the distance traveled divided by the time taken. The question asks for