# Review of Linear Motion

## Alexei Gilchrist

### 1 Kinematics

The basic kinematic variable for an object is its position \(x\). In two or more dimensions this is a vector, e.g.

Note that the basis, or co-ordinate system you choose is just that—a choice. There are an infinite number of possible bases and they don’t even have to be an orthogonal set, though this usually makes calculations much easier. The basis elements just have to be linearly independent and span the space (that is, you can get to any point by some linear combination of basis elements). You should choose whatever basis makes the problem at hand look the simplest. We’ll almost always choose an orthonormal basis (orthogonal and unit length basis vectors).

The length of the position vector (distance from the origin) is \(\sqrt{\vec{r}.\vec{r}} = r\), and if the basis is orthonormal it’s just the square root of the the sum of the squares of the components, \(r=\sqrt{x^2+y^2+z^2}\).

The instantaneous change in time of the position gives the velocity:

The final kinematic variable is the acceleration which is the change in the velocity with time

### 2 Dynamics

The various time derivatives of position give a description of *how* an object moves with time, usually expressed as an equation of motion. *Why* an object should move in a particular way is the subject of dynamics.

The next quantity to add to our list is the mass of an object, \(m\). Essentially this captures an objects resistance to changes in velocity. This relationship is expressed neatly by Newton’s second law:

Rewriting Newton’s second law slightly,

Finally we come to energy \(E\), where we distinguish between two types: kinetic \(T\) and potential \(U\). The kinetic energy is energy associated with motion:

For a conservative system the potential energy \(U\) determines the force:

Given this relationship between the potential energy and the force, it is tempting to plot the potential energy and imagine a small ball rolling on top of this potential energy surface (or a curve in 1-D). Certainly this analogy reproduces some of the behavior of an object moving under such a potential. For instance the direction of motion is correct—the ball will roll downhill which corresponds to a force in the opposite direction to the slope. The ball will sit still at points where the surface is flat and indeed there is no force on the object at this point as \(dU/dx=0\), and you can even tell if this equilibrium point is stable or unstable by considering what will happen if you give the ball a little nudge. That increasing slope corresponds to increasing force is also correct. However it’s possible to take this analogy too far. For instance, the velocity of the ball has no connection to the object. A vertical wall in the potential would correspond to an infinite force, but for the ball there would be no change in position (motion downwards has no meaning in the analogy). It’s a great analogy, just be mindful of it’s limitations.