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In projectile motion, what does it mean for the motion along the $x$ and $y$ axis to be independent?

This question is referring to the concept of projectile motions.

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We just calculate the motion along the x and y axis separately to make the calculation easier. Since acceleration due to gravity only acts along the y axis, when we seperate the calculation for x and y axis then in the x axis calculation we just say 'it's moving at constant velocity' whereas in y axis we just say 'it's accelerating downwards'.

Also, we can calculate stuff like variation of distance and height with time.

With more advanced math you can do cool stuff like 'rotating' and 'moving' the x and y axes.

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The position of a projectile in two dimensions at a time $t$ can be specified by a pair of numbers $(x(t), y(t))$ giving its $x$ and $y$ positions at time $t$. In projectile motion, Newton's second law tells us that the equation of motion for the projectile can be written in terms of the two functions $x(t)$ and $y(t)$ as follows: $$ \ddot x(t) = 0, \qquad \ddot y(t) = -g $$ where overdots denote time derivatives. Here we have assumed that the $x$-axis is horizontal (parallel to the Earth's surface), and the $y$ axis is vertical with the positive direction pointing away from the Earth's surface, and $g$ is the magnitude of the acceleration due to gravity.

Here's the main point about "independence of the $x$ and $y$ motion" in this case. Notice that the equation for $x(t)$ does not involve $y(t)$, and the equation for $y(t)$ does not involve $x(t)$. The technical term for this is that the $x$ and $y$ equations are uncoupled differential equations. So we can solve for $x(t)$ without knowing anything about $y(t)$ and vice versa. We can therefore solve the two equations of motion independently of one another to obtain the general solutions $$ x(t) = x(0) +\dot x(0) t, \qquad y(t) = y(0) +\dot y(0) t - \frac{1}{2} gt^2 $$

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It means that all kinematical relations to describe one-dimensional motion along X-axis can't influence motion along Y-axis and vice versa.

For example, if you're applying force along X-axis only, any component of acceleration can't be along Y-axis. The force can cause acceleration only along X-axis.
In more general, if you're applying force an arbitrary direction, only X-component of force can cause acceleration along X-axis and only Y-component of force can cause acceleration along Y-axis.

Simplest example: If you are running from North to South, you can't be displaced towards East or West.

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