08Fkinematics

= 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 often 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 the length of the entire journey the particle has been on.


 * **Displacement** is the length between where the particle finishes compared to where it started and needs a direction as well as length.


 * **Average Speed** is distance travelled divided by time.


 * **Velocity** is the __rate of change__ of __position__ with respect to __time__.
 * so ** Average Velocity ** is __**displacement**__ divided by time


 * **At rest** means speed = 0


 * ** Initial ** means time = 0
 * We don't do negative time, so time will always start at 0


 * Rectilinear Motion **


 * ** Rectilinear Motion ** is movement back and forth along a straight line


 * Examples of rectilinear motion are
 * a paddle in the old computer game of pong (or the ship in space invaders)
 * sprinters in a 100m race
 * people in a lift


 * In this course, we will only study rectilinear motion


 * Position-Time Graphs **


 * A graph with time elapsed on the horizontal axis and position of the object on the vertical axis


 * Useful for analysing the motion of an object (especially rectilinear motion)


 * Position can be positive, zero or negative


 * We don't do negative time, so position-time graphs will always only have positive values on the horizontal axis


 * Example 1 **


 * Benny was sitting on a ramp, playing with his toy car. By a fortunate coincidence, the ramp is marked out with a scale.


 * Benny pushed his car up the ramp, letting it go at the +2 marker (at t = 0). After 1 second, the car passed the +5 marker but it was slowing down The highest point it reached was the +6 marker (at t = 2)


 * Then the car started rolling back down the ramp towards Benny


 * At this point, Benny was distracted by the crazed maths teacher who was watching excitedly while holding a notepad and a stopwatch. As a result, Benny missed catching his car and it zoomed past him and on down the ramp in a valiant bid for freedom.


 * The crazed maths teacher drew the number line to the right to record the journey of the car.


 * 1) Find the distance travelled by the toy car in the first five seconds (t = 0 to t = 5)
 * 2) Find the average speed during the first five seconds
 * 3) Find the displacement of the car after five seconds
 * 4) Find the average velocity of the car during the first five seconds
 * 5) Draw a position-time graph for the car
 * 6) In which part of the graph was the velocity positive?


 * Solution:**

... ... ** 1. ** . ** Find the distance travelled by the car in the first five seconds (t = 0 to t = 5) **

... ... ... The car travels up the ramp 4 units ... ... ... and down the ramp another 9 units

... ... ... so total distance travelled = 13 //units//

... ... ** 2. ** . ** Find the average speed during the first five seconds **

math . \qquad \quad \text{Average Speed } = \dfrac{distance}{time} \\.\\ . \qquad \qquad \qquad \qquad \qquad = \dfrac{13}{5} \\.\\ . \qquad \qquad \qquad \qquad \qquad = 2.6 \; units/second math

... ... ** 3. ** . Find the displacement of the car after five seconds

... ... ... Displacement = Final Position – Initial Position

... ... ... Displacement = –3 – 2

... ... ... Displacement = –5 //units//

... ... ** 4. ** . ** Find the average velocity of the car during the first five seconds **

math . \qquad \text{Average Velocity } = \dfrac{displacement}{time} \\.\\ . \qquad \qquad \qquad \qquad \qquad = \dfrac{-5}{5} \\.\\ . \qquad \qquad \qquad \qquad \qquad = -1 \; units/second math

... ... ** 5. ** . ** Draw a position-time graph for the car **

... ... ... See graph

... ... ... Note that the known points have been joined with a smooth curve

... ... ** 6. ** . ** In which part of the graph was the velocity positive **

... ... ... The velocity is positive when the graph is sloping up

... ... ... The velocity is positive when 0 __<__ t < 2

... ... ... At t = 2, the instantaneous velocity is 0

... ... ... The velocity is negative when t > 2


 * Velocity-Time Graphs **


 * A graph with time elapsed on the horizontal axis and velocity on the vertical axis
 * Also useful for studying the motion of an object


 * Velocity can be positive, zero or negative


 * A position-time graph that slopes up has positive velocity
 * A position-time graph that is horizontal has zero velocity
 * A position-time graph that slopes down has negative velocity


 * A position-time graph that is a straight line will have a constant velocity
 * so the velocity-time graph will be a horizontal line


 * Example 2 **


 * Given the the crazed maths teacher measured
 * the initial velocity for Benny's car as 4 units/second
 * the velocity at t = 2 was 0 units/second
 * Draw a Velocity-Time graph for Benny's car from Example 1


 * Solution:**


 * see graph


 * notice that
 * velocity is positive (but decreasing) for time < 2
 * the car going up the ramp but slowing down
 * velocity is zero at time = 2
 * the car stops for an instant
 * velocity is negative for time > 2
 * the car starts rolling down the ramp
 * the magnitude of the velocity is increasing for t > 2
 * as the car speeds down the ramp


 * Example 3 **


 * Benny leaves his home and walks to the shop at a constant pace of 20 meters/minute
 * The shop is 200m from his house
 * assume Benny's house is at zero
 * Benny buys an icecream at the shop and sits outside the shop for 10 minutes while he eats the icecream
 * Then Benny notices the crazed maths teacher lurking behind a tree and holding a notepad and a stopwatch.
 * Benny starts running home, getting faster and faster in an attempt to get away from the crazed maths teacher


 * 1) Draw a Position-Time graph of Benny's trip
 * 2) Draw a Velocity-Time graph of Benny's trip


 * Solution:**





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