# What does it mean to find an equation of motion, given vector functions that describe both the object's position and velocity?

I don't really understand how to approach a problem that asks to find the equation of motion. Intuitively, I would guess that an "equation of motion" is an equation where the particle's position is given as a function of some independent variable(s). But doesn't this describe the vector function of the position, namely $\vec{r(t)}$?

I don't understand what this equation of motion would look like. Therefore, I have no target in mind so I'm hopelessly lost. To me, it seems sufficient to have the position vector function (which will tell me where the object is at any time, $t$) and the velocity vector function (which tells me its velocity at any point).

For example, if I'm given that the position of a small bead of mass $m$ is given by $\vec{r(\theta)}=(R\cos \theta, R \sin \theta, q \theta)$ I can find the velocity by taking the derivative (assume there is also a gravitational acceleration in the $-z$ direction). I can also find the energies and whatnot. Then, what would the equation of motion look like?

Update: going off of the comments/answers, I derived the equation for the acceleration to be

$$a=-\frac{gq}{R^2+q^2}$$

So now technically I've found all of the equations for position, velocity, acceleration, kinetic energy, potential energy, and total energy....

• looking for other people to confirm, but this sounds like you're looking for the SUVAT equations? – Alex Robinson Oct 16 '16 at 19:07
• @Cursed1701 Yes, these are actually called equations of motion. But I have, as the OP, often run into sentences mentioning the equation of motion. It is a way to talk about that one specific equation which can describe the whole motion in one - there might be more that can do that, and so each of those is an equation of motion. – Steeven Oct 16 '16 at 19:18

## 1 Answer

An equation of motion of an object is an equation that describes the entire motion in time (with time $t$ as the only unknown).

As you rightfully say, if you have the position expression (as a function of time), you can find the velocity and acceleration expressions ect. And reverse; if you have the velocity expression, you can find the acceleration as well as the position equation (in the latter case you must be given some starting conditions for the integral).

And if that is the case - if you have an equation from which you can derive each other relevant motion parameter as function of time - then you have what you are looking for.

• Well, then I am entirely confused as to what this question wants. I'm given $\vec{r(\theta)}$, $\theta=\theta (t)$, and I've found $\vec{v(\theta)}$, kinetic energy, potential energy, and total energy. Aside from finding the acceleration, what does it mean to ask for an equation of motion, given that it is part E of a multi-part question? I guess they insist on there being another equation that I need to look for, but I have no clue what else is left to find! – whatwhatwhat Oct 16 '16 at 19:07
• @Sometimes - I have often run into it in oscillation classes - it is refering to a specific setup of an equation. An example I could dig out quickly is a spring bouncing back and forth: $$kx=-m\ddot{x}$$ $k$ and $m$ are known values, so this is an equation that describes the motion completely. – Steeven Oct 16 '16 at 19:16
• The simplest EoM is that of a body falling in a central gravitational field, with no other forces acting: $m\ddot{x}=mg$. With initial conditions $\dot{x(0)}=v_0$ and $x(0)=x_0$, these fully describe the motion of the object. – Gert Oct 16 '16 at 19:20
• Hmmm...your responses are pointing me in the direction of finding the equation for the object's acceleration., which kinda makes sense because I have not yet found that. But I should point out that this is helical motion. – whatwhatwhat Oct 16 '16 at 20:46