Speed of Object Approaching Convex Mirror

To find the speed of the object, we first calculate its initial and final positions using the mirror formula.

1. Focal length of the convex mirror:
The focal length $f$ of a convex mirror is positive and is related to its radius of curvature $R$ by:
$f = +\frac{R}{2} = +\frac{20\text{ m}}{2} = +10\text{ m}$

2. Mirror Formula:
$\frac{1}{f} = \frac{1}{v} + \frac{1}{u} \implies \frac{1}{u} = \frac{1}{f} - \frac{1}{v}$

3. Initial position of the object ($u_1$):
Given the initial image distance $v_1 = +\frac{25}{3}\text{ m}$:
$\frac{1}{u_1} = \frac{1}{10} - \frac{1}{\frac{25}{3}} = \frac{1}{10} - \frac{3}{25}$
$\frac{1}{u_1} = \frac{5 - 6}{50} = -\frac{1}{50}\text{ m}^{-1}$
$u_1 = -50\text{ m}$

Thus, the initial distance of the object from the mirror is $50\text{ m}$.

4. Final position of the object ($u_2$):
Given the final image distance $v_2 = +\frac{50}{7}\text{ m}$:
$\frac{1}{u_2} = \frac{1}{10} - \frac{1}{\frac{50}{7}} = \frac{1}{10} - \frac{7}{50}$
$\frac{1}{u_2} = \frac{5 - 7}{50} = -\frac{2}{50} = -\frac{1}{25}\text{ m}^{-1}$
$u_2 = -25\text{ m}$

Thus, the final distance of the object from the mirror is $25\text{ m}$.

5. Speed of the object:
The distance traveled by the object in time $\Delta t = 30\text{ s}$ is:
$\Delta x = |u_2 - u_1| = |-25 - (-50)| = 25\text{ m}$

The speed of the object in meters per second ($\text{m/s}$) is:
$v_{\text{obj}} = \frac{\Delta x}{\Delta t} = \frac{25\text{ m}}{30\text{ s}} = \frac{5}{6}\text{ m/s}$

Converting this speed to $\text{km/h}$:
$v_{\text{obj}} = \frac{5}{6} \times \frac{18}{5}\text{ km/h} = 3\text{ km/h}$

The speed of the object is $3\text{ km per hour}$.