# Motion in a straight line

## Basic Definitions

### Position

First, we will talk about one of the most basic concepts of physics, position. There are many ways to express the position of a point particle, one of the simplest and most well known way being the Cartesian Coordinate System. In a Cartesian Coordinate System, the position of a particle is expressed as an ordered pair, the x-coordinate, a y-coordinate in a two or three dimensional space, and a z-coordinate in a three dimensional space.

${\displaystyle P=(x,y,z)\,}$

The point (0, 0, 0) is commonly known as the origin. The x, y, and z axes pass through the origin at right angles to create the foundation for point-plotting.

### Displacement

${\displaystyle \mathbf {d} =\Delta \mathbf {r} =\mathbf {r} _{2}-\mathbf {r} _{1}}$

### Velocity

The velocity is defined to be the rate of change of position with respect to time. For now, this basic definition should hold. Just knowing the basic definition of velocity should be enough to answer many of the problems we will investigate later. The SI unit for velocity is the meter/second. This should not surprise you as the SI unit for length is the meter and the SI unit for time is the second.

Since velocity is the rate of change of position, one can conclude that

${\displaystyle \Delta x=vt\,}$

And since

${\displaystyle \Delta x=x_{f}-x_{o}\,}$
${\displaystyle x_{f}=x_{o}+vt\,}$

We have just derived a basic formula to calculate the position of an object given the initial position, a constant velocity, and the time elapsed.

There is a significant difference between velocity and speed. Velocity is a vector - It has two pieces of information: a magnitude and a direction. Speed is the magnitude of velocity, a scalar value. An important example of this distinction is centripetal acceleration. Since the definition of acceleration is a change in velocity, there can be a change in speed, direction or both. In centripetal acceleration, the speed of a particle in uniform circular motion (such as a weight attached to a string swung above one's head) does not have to change, though since its direction is constantly changing it experiences acceleration.

#### Examples

1. If point x has an initial position at 0 and travelling with constant velocity 3 m/s, find the position at 7 sec. Answer: To solve this problem, simply use the formula we derived earlier

${\displaystyle x_{f}=x_{o}+vt\,}$
${\displaystyle x_{f}=0m+3m/s(7sec)\,}$
${\displaystyle x_{f}=21m\,}$

### Speed

Speed is the magnitude of velocity. It is a scalar quantity (i.e., not a vector). The direction of the particle does not influence its speed. For example, a particle traveling east at 10 meters per second has the same speed as a particle traveling west 10 meters per second; however, these two particles would have opposite velocities because they are traveling in opposite directions.

### Acceleration

Acceleration is the rate of change of velocity. It is the derivative of velocity over time, as well as the second derivative of position over time.

### Momentum

Momentum is mass times velocity. Momentum is always conserved, this is known as Conservation of Momentum.

${\displaystyle \mathbf {p} =m\mathbf {v} \,}$

### Force

According to Newton's Second Law of Motion, force is mass times acceleration. Force is commonly confused with momentum. The SI unit of force is the newton.

${\displaystyle F=ma\,}$

Suppose a car with a mass of 200kg is accelerating at 5 m/s/s. Since F = ma, you can calculate the force to be 1000 Newtons

However, if same car had a constant velocity of 5 m/s, the force is 0N because the car is not accelerating.

## Simple Forms of Motion

### Constant Velocity

${\displaystyle v=(x_{f}-x_{o})/t\,}$

### Constant Acceleration

${\displaystyle a=a(t)=a_{0}\,}$
${\displaystyle v=v(t)=v_{0}+a_{0}t\,}$
${\displaystyle x=x(t)={1 \over 2}(v_{0}+v(t))t\,}$

${\displaystyle x=x(t)=x_{0}+v_{0}t+{1 \over 2}a_{0}t^{2}\,}$