As a projectile of mass $m$ kilograms travels through air, it experiences a frictional force - HSC - SSCE Mathematics Extension 2 - Question 8 - 2022 - Paper 1

According to Newton's second law, the net force $F$ is equal to the mass $m$ times the acceleration $a$:

$F = m a$.

In this case, the net force acting on the projectile is given by the difference between the gravitational force and the frictional force.

The frictional force is proportional to the square of the speed, thus we can express the frictional force as:

$F_{friction} = -k v^2$,

where $k$ is a positive constant.

Combining these forces, we have:

$-mg - kv^2 = m a$,

which leads to:

$m a = -mg - kv^2$.

This correctly accounts for both gravitational and frictional forces acting on the projectile.

Based on the derived equation, the correct model of the projectile's motion corresponds to option B:

$(-mg) - kv = m \frac{d^2 \bold{r}}{dt^2}$.