1. Use Einstein’s photoelectric equation

For incident photon energy $E$ and work function $\phi$,

$K_{\max} = E - \phi$

and

$K_{\max} = \frac{1}{2}mv_{\max}^2$

So the maximum speed is proportional to the square root of the maximum kinetic energy:

$v_{\max} \propto \sqrt{K_{\max}}$

2. Case 1: Photon energy = $5\phi$

Then

$K_1 = 5\phi - \phi = 4\phi$

So,

$v_1 \propto \sqrt{4\phi} = 2\sqrt{\phi}$

3. Case 2: Photon energy = $10\phi$

Then

$K_2 = 10\phi - \phi = 9\phi$

So,

$v_2 \propto \sqrt{9\phi} = 3\sqrt{\phi}$

4. Find the ratio of maximum velocities

$v_1 : v_2 = 2\sqrt{\phi} : 3\sqrt{\phi} = 2:3$

5. Evaluate options

• A: $1:2$ ❌
• B: $1:3$ ❌
• C: $2:3$ ✅
• D: $3:2$ ❌

Thus, the correct option is C.