A body is projected from the ground with some angle to the horizontal. What happens to the angular momentum about the initial position in this motion

To determine what happens to the angular momentum of a projectile about its initial position, we can analyze the definition of angular momentum and the torque acting on the projectile.

Let us consider a body projected from the ground (origin, $O$) with an initial velocity $u$ at an angle $\theta$ to the horizontal.
The angular momentum $L$ of the body of mass $m$ about the initial position (origin) is defined as:
$\overrightarrow{L}=\overrightarrow{r}\times \overrightarrow{p}$
where $\overrightarrow{r}$ is the position vector of the projectile relative to the origin and $\overrightarrow{p}$ is its linear momentum ($\overrightarrow{p}=m\overrightarrow{v}$).

Alternatively, the rate of change of angular momentum about the origin is equal to the net external torque $\overrightarrow{\tau }$ acting on the body about the origin:
$\frac{d\overrightarrow{L}}{dt}=\overrightarrow{\tau }$

During the projectile motion, the only force acting on the body (neglecting air resistance) is gravity, which acts vertically downwards:
$\overrightarrow{F}=-mg\hat{j}$

Let the coordinates of the projectile at any time $t$ be $(x,y)$. Thus, the position vector is:
$\overrightarrow{r}=x\hat{i}+y\hat{j}$

The torque $\overrightarrow{\tau }$ about the origin $O$ is given by:
$\overrightarrow{\tau }=\overrightarrow{r}\times \overrightarrow{F}=(x\hat{i}+y\hat{j})\times (-mg\hat{j})$
Since $\hat{i}\times \hat{j}=\hat{k}$ and $\hat{j}\times \hat{j}=0$, we have:
$\overrightarrow{\tau }=-mgx\hat{k}$

The horizontal distance $x$ is always positive throughout the flight because the projectile moves in the positive x-direction from the start:
$x=(ucos\theta )t$

Thus, the magnitude of the torque is:
$\tau =mgx=mgucos\theta t$

Since $\tau =\frac{dL}{dt}$, the rate of change of angular momentum is directly proportional to time $t$ and is always positive:
$\frac{dL}{dt}>0$ (for $t>0$)

This indicates that the angular momentum of the body about the initial point increases continuously with time throughout the motion.

Therefore, the angular momentum about the initial position in this motion increases.