Solution

Statement

$2.2.11.$ A proton with an initial velocity $v$ flies directly at an initially stationary helium nucleus. What is the velocity of the particles at their closest approach? The mass of the helium nucleus is close to four times the mass of the proton.

Solution

At the closest approach, the relative velocity of the atom and the proton will be zero, meaning their absolute velocities are equal. Let's write the law of conservation of momentum:
$$mv = mv_1 + 4mv_2$$
$$v_1 = v_2$$

From this:
$$mv = 5mv_1 \implies v_1 = v/5$$