Momentum, collisions and motion under a variable force

Learning objectives

Analyse the oblique impact of two smooth spheres by resolving along and perpendicular to the line of centres, keeping the perpendicular components unchanged and applying momentum and Newton’s experimental law along the line

Apply conservation of linear momentum to direct and oblique impacts between two particles, together with Newton's experimental law in the form e = speed of separation / speed of approach, where 0 ≤ e ≤ 1

Calculate the kinetic energy lost in an impact, and explain why no energy is lost when e = 1, why the loss is greatest when e = 0 and the particles coalesce, and why kinetic energy can never increase

Analyse the oblique impact of a smooth sphere with a fixed plane by leaving the component of velocity parallel to the plane unchanged and reversing and multiplying the perpendicular component by e

Form and solve the equation of motion for a particle under a variable force, choosing a = dv/dt when the force depends on time and a = v dv/dx when it depends on displacement, and separating the variables

Model resisted motion with resistance proportional to v or to v², find the terminal speed by setting the acceleration to zero, and determine the distance or time to reach a given speed by integration