Tag Info

The description of the movement of bodies by their position, velocity, acceleration (and possibly higher time derivatives, such as, jerk) without concern for the underlying dynamics/forces/causes.

## When to Use this Tag

Use to discuss the movement of a body in terms of position, velocity, acceleration (or, in principle, higher derivatives thereof, such as, jerk) without concern for the forces/dynamics causing this movement.

## Introduction

The classical description of the movement of a (point-like) body consists of three vector quantities, defined in a suitable background coordinate system (usually $\mathbb{R}^n$ for n-dimensional problems).

• The position of the body, usually denoted by either $\vec x(t)$ or $\vec q(t)$ as a function of the time $t$.
• The first total time derivative of the position of the body, defined to be the velocity $v(t) \equiv \frac{\mathrm{d}\vec x(t)}{\mathrm{d}t}$.
• The second total time derivative of the position of the body, defined to be the acceleration $a(t) \equiv \frac{\mathrm{d}\vec v(t)}{\mathrm{d}t} = \frac{\mathrm{d}^2\vec x(t)}{\mathrm{d}t^2}$.

## Special Cases

### Constant Velocity

Problems in which some body travels with a constant velocity are common introductory exercises and can be solved with the difference version of the definition of velocity:

$$\vec v = \frac{\Delta \vec x}{\Delta t} = \frac{\vec x - \vec x_0}{t - 0}\quad,$$

where we take the body to be at position $x_0$ at time $t = 0$.

### Constant acceleration

In some problems, the acceleration of the body is a constant $\vec a_0$, for example $\vec g$ during a free fall close to the surface of Earth. In this case, it is easy to integrate twice to calculate the position $\vec x$. With initial conditions $\vec x(0) = \vec x_0$ and $\vec v(0) = \vec v_0$, we have:

\begin{eqnarray} \vec a(t) & = & \vec a_0 \\ \vec v(t) & = & \vec a_0 t + \vec v_0 \\ \vec x(t) & = & \frac{1}{2} \vec a_0 t^2 + \vec v_0 t + \vec x_0 \end{eqnarray}

### Constant Jolt

, or is the rate of change of acceleration with respect to time; i.e. $\vec j=\frac{\mbox{d}\vec a}{\mbox{d}t}$. In the case of a constant jolt, one may trivially apply the Taylor expansion (or through algebraic means) to find that:

\begin{eqnarray} \vec j(t) & = & \vec j_0 \\ \vec a(t) & = & \vec j_0 t + \vec a_0 \\ \vec v(t) & = & \frac{1}{2} \vec j_0 t^2 + \vec a_0 t + \vec v_0 \\ \vec x(t) & = & \frac{1}{6} \vec j_0 t^3 + \frac{1}{2} \vec a_0 t^2 + \vec v_0 t + \vec x_0 \end{eqnarray}