# How can I calculate the speed of an object knowing its horizontal and vertical velocity components?

Let's say a ball is thrown and it experiences typical projectile motion (moves in a parabolic arc etc.) and the only information we know are the equations for the horizontal and vertical components of its velocity for it's entire path.

From the given information, how does one calculate the total/actual speed of the ball relative to the direction it is travelling in at any given point (ignoring drag)?

As an example (horizontal and vertical components of velocity respectively):

$$V_x = 30$$

$$V_y = 20 - 9.81t$$

Is it simply a matter of using Pythagoras' theorem and taking the magnitude?

$$V=\sqrt{(V_x)^2 + (V_y)^2}$$

The formula you have written is correct; but they are functions of time. Hence, by inserting the particular instant , say $t$ on the function ,you get the instantaneous components of velocity. Then using phythagoras theorem you will get the total instantaneous velocity. Taking your example, at time $T$ s , the X-comp. is $30$ unit and Y-comp. is $(20 - 9.81T)$ unit . So,velocity at time $T$ is $\sqrt{(30)^2 + (20 - 9.81T)^2}$ unit
Yes, of course. Velocity is just a vector, and its norm is $\sqrt{V \cdot V}$, i.e., $\sqrt{\sum_i V_i ^2}$