IB Physics Air Resistance is Small (but not negligible)

Most textbook problems state that air resistance, the drag force on a projectile due to friction as it moves through a fluid medium, is to be neglected. A tutorial sheet of “show that” questions for the case where the drag force is small compared to the weight are given below. All questions give approximate solutions, correct to the first order in k/m, where k is a small positive number called the drag constant and m is the mass of the particle.

1. A particle of mass m falls vertically from rest through air. If the drag force is given by f_d = -k v, where v is the speed of the object, show that the distance fallen at time t is given by s = 1/2 gt²( 1 - kt/(3m) ).
2. A particle of mass m falls vertically from rest through air. If the drag force is given by f_d = -k v², where v is the speed of the object, show that the distance fallen at time t is given by s = 1/2 gt²( 1 - gkt²/(6m) ).
3. A particle of mass m is projected at a speed u at an angle 𝛼 to the horizontal in air where the drag force is proportional to the velocity vector of the particle, f_d = -k v⃗, where k is a small positive constant. Show that the time taken to reach the highest point is given by t = u sin𝛼/g - k u²sin²𝛼/(2mg²).
4. A particle of mass m is projected at a speed u at an angle 𝛼 to the horizontal in air where the drag force is proportional to the velocity vector of the particle, f_d = -k v⃗, where k is a small positive constant. Show that the path of the projectile is y = x tan𝛼 - g x²sec²𝛼/(2u²) + k/m( x²sin𝛼/(2ucos²𝛼) - 2/3x³/(ucos𝛼)³ )