2.3.35. A particle of mass $m$ flies into a region where it is affected by a braking force that depends only on the distance between the particle and the boundary of the region. Find this dependence if the depth of penetration of a particle into the braking region is proportional to its initial momentum: $l = \alpha p$.
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} \quad dA = dE \Rightarrow F dl = \frac{p dp}{m} \$$
