Solution

The work done by the force $F$ when the particle moves $dl$

$$F dl = dA \tag{1}$$

From the law of conservation of energy

$$\quad \frac{m u^2}{2} = \frac{p^2}{2m}$$

Considering a small increment of energy

$$dE = \frac{d(p^2)}{2m} = \frac{p dp}{m} \tag{2}$$

$$dA = dE \Rightarrow F dl = \frac{p dp}{m}$$

$$\quad F = \frac{p}{m} \frac{dp}{dl} = \frac{p}{m} \frac{1}{\alpha} = \frac{1}{m} \frac{l}{\alpha} \frac{1}{\alpha} = \frac{l}{m \alpha^2} \quad (*)$$

From where

$$\quad \frac{dp}{dl} = \frac{1}{\alpha} , \quad p = \frac{l}{\alpha} \\ \quad \boxed{F = \frac{l}{m \alpha^2}}$$