8AKinematicsIntro


 * Introduction to Kinematics**


 * **Kinematics** is the study of motion (movement) without considering why the motion is occurring.
 * The study of the forces that cause motion is called ** mechanics **


 * The word **particle** (or **body** ) is used to describe the object that is moving.


 * We treat the particle/body as if it occupies a single point in space.
 * The dimensions of the body are not important.
 * We don't consider anything going on within the body -- eg spin or stretch or compression.

Definitions


 * ** Position ** is the location of the particle compared to a fixed point (often, but not always, the starting point).


 * **Distance** (or **Distance Travelled** ) is a scalar (size) and is the length of the total journey of the particle.


 * **Displacement** is a vector (direction and size) and is the distance between where the particle finishes compared to where it started.


 * **Average Speed** is a scalar (size) and is distance travelled divided by time.


 * **Velocity** is a vector (direction and size) and is the __ rate of change __ of __ position __ with respect to __ time __.
 * ** Average Velocity ** is the Displacement divided by time.


 * **Speed** (at time t) is the magnitude of velocity at time t.


 * **At rest** means speed = 0


 * **Acceleration** is a vector (direction and size) and is the __ rate of change __ of __ velocity __ with respect to __ time __.


 * **Deceleration** or **retardation** are both used to describe slowing down (negative acceleration).

Notation


 * **Position** at time t is given by ** x(t) ** or sometimes ** s(t) **


 * **Velocity** at time t is given by ** v(t) **

math . \qquad \qquad v(t)=\dot{x}(t)=\dfrac{dx}{dt} math


 * **Acceleration** at time t is given by ** a(t) **

math \\ . \qquad \qquad a(t)=\dot{v}(t)=\dfrac{dv}{dt} \\. \\ . \qquad \qquad a(t)=\ddot{x}(t)=\dfrac{d^2x}{dt^2} math

Rectilinear Motion


 * **Rectilinear motion** is movement in 1 Dimension
 * ie backwards and forwards along a straight line

Example

A particle moves in a straight line so that its position, x metres from O, at time t seconds, is given by: math . \qquad x(t)=5+4t-t^2 math

a) Find the particle's initial position


 * when t = 0, x(0) = +5m
 * Always assume positive x is to the right of O

b) Find x after 2 seconds


 * when t = 2, x(2) = +9m

c) Find the initial velocity

math . \qquad \text{Velocity is } \; v(t)=\dot{x}(t)=4-2t math


 * when t = 0, v(0) = +4m/s

d) Find the velocity after 2 seconds


 * when t = 2, v(2) = 0m/s
 * ie instantaneously at rest

e) Find the initial acceleration

math . \qquad \text{Acceleration is } \; a(t)=\dot{v}(t)=-2 math


 * a(0) = –2 m/s 2
 * Acceleration is constant
 * Acceleration in opposite direction to velocity so particle is slowing down

f) Find the acceleration after 2 seconds


 * a(2) = –2 m/s 2
 * Instantaneously at rest but accelerating back towards the origin

g) Find the position and velocity after 3 seconds


 * x(3) = +8m
 * Has now moved back towards origin


 * v(3) = –2 m/s
 * acceleration in same direction as velocity so speeding up in negative direction

h) Find the displacement and distance travelled in the first 3 seconds


 * ** Displacement **
 * ** = ** x(3) – x(0)
 * = 8 – 5
 * = +3m


 * ** Distance travelled **

.
 * ==> travelled from x = 5 to x = 9 then to x = 8,
 * = 4 + 1
 * = 5m