06 8kinematics

Kinematics

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Kinematics is the study of the motion of a particle.

  • In Kinematics, we are not concerned with the forces that make that particle move.
  • Because we are simplifying to a single particle, we are not concerned with the object stretching, compressing, spinning or changing shape in any other way.

For this course, we are only considering motion backward and forward along a straight line.

  • Motion along a straight line is sometimes called Rectilinear Motion.

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Definitions

Particle or Object refers to the thing that is moving. A particle has a length of zero. In this course, we model all moving objects as particles.

Position refers to where the particle is placed on the straight line compared to a fixed "origin"

  • The origin may or may not be the original position of the particle.
  • Position has direction so it can be either positive or negative.
  • We often use the function $x(t)$ to refer to the position at time, t.

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Displacement is the difference in position between where the particle started and where it is now.

  • Displacement has direction so it can be either positive or negative.
  • Displacement between t1 and t2 is given by $d(t) = x(t_2) - x(t_1)$

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Distance Travelled refers to the journey the particle has taken to get where it is now.

  • Distance travelled does not have direction, it can only be a positive value.

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  • Position, Displacement and Distance Travelled are measurements of distance so they will commonly be measured in cm, m, km, etc

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Velocity is the rate of change of position. It is equal to the displacement divided by the time.

  • Velocity does have direction.
  • Average Velocity $= \dfrac{\text{change in position}}{\text{change in time}} = \dfrac{x(t_2) - x(t_1)}{t_2 - t_1}$
  • Instantaneous Velocity $= \dfrac{dx}{dt}$ … rate of change of position with respect to time.

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Speed is equal to the distance travelled divided by the time.

  • Speed does not have direction.
  • Average Speed $= \dfrac{\text{distance travelled}}{\text{change in time}}$
  • Instantaneous Speed = Magnitude of Instantaneous Velocity

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  • Speed and Velocity are rates of change of position with respect to time. They will commonly be measured in m/s, km/h, etc

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Acceleration is the rate of change of velocity. It is equal to the change in velocity divided by time.

  • Acceleration has direction.
  • Accleration is a rate of change with respect to time. It will commonly be measured in m/s/s (or m/s2), etc.
  • Average Acceleration $= \dfrac{\text{change in velocity}}{\text{change in time}} = \dfrac{v(t_2) - v(t_1)}{t_2 - t_1}$
  • Instantaneous Acceleration $= \dfrac{dv}{dt} = \dfrac{d^2x}{dt^2}$ … rate of change of velocity with respect to time

Other common phrases

Momentarily at rest means that the object briefly comes to a stop before moving off again.

  • If the object is momentarily at rest, then the velocity is zero $\dfrac{dx}{dt} = 0$

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Example 1

A particle moves in a straight line such that its position is given by:
… … $x(t) = t^2 - 4t + 3, \qquad t \geqslant 0$

… … where x is measured in metres and t is measured in seconds

a) .. Find the original position
b) .. Find the original velocity and describe the initial direction of motion
c) .. Find the original acceleration
d) .. Find the time and position when the particle is momentarily at rest
e) .. Find the position, and velocity after 3 seconds
f) .. Find the average velocity and average speed over the first three seconds
g) .. Draw a Position-Time Graph
h) .. Draw a Velocity-Time Graph

Solution

a) .. Find the original position

… … … $x(t) = t^2 - 4t + 3$

… … … $x(0) = 3$

… … … initial position at +3m (or 3m right of origin)
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b) .. Find the original velocity and describe the initial direction of motion

… … … $x(t) = t^2 - 4t + 3$

… … … $v(t) = x'(t) = 2t - 4$

… … … $v(0) = -4$

… … … original velocity is -4m/s (or a speed of 4 m/s to the left)
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c) .. Find the original acceleration

… … … $v(t) = 2t - 4$

… … … $a(t) = v'(t) = 2$

… … … $a(0) = 2$

… … … acceleration is a constant 2 m/s2 to the right
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d) .. Find the time and position when the particle is momentarily at rest

… … … $v(t) = 2t - 4$

… … … momentarily at rest when $v(t) = 0$

… … … $2t - 4 = 0$

… … … $t = 2$
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… … … $x(t) = t^2 - 4t + 3$

… … … $x(2) = -1$

… … … particle is at rest at 2 seconds and is at -1m (1m left of origin)
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e) .. Find the position, and velocity after 3 seconds

… … … $x(t) = t^2 - 4t + 3$

… … … $x(3) = 0$
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… … … $v(t) = 2t - 4$

… … … $v(3) = 2$

… … … At 3 seconds, particle is at the origin (x = 0) and has velocity 2 m/s to the right
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f) .. Find the average velocity and average speed over the first three seconds

… … … In first three seconds so between $t = 0 \text{ and } t = 3$

… … … Average Velocity

… … … … $= \dfrac{\text{change in position}}{\text{change in time}}$

… … … … $= \dfrac{x(3) - x(0)}{3 - 0}$

… … … … $= \dfrac{0 - 3}{3}$

… … … … $= -1$ m /s
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… … … Average Speed

… … … … $= \dfrac{\text{distance travelled}}{\text{change in time}}$

… … … … $= \dfrac{\text{distance from 3 to -1 then from -1 to 0}}{3 - 0}$

… … … … $= \dfrac{4 + 1}{3}$

… … … … $= \dfrac{5}{3}$ m/s
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g) .. Draw a Position-Time Graph

… … … A Position-Time Graph always has:

… … … … Time on the horizontal axis.

… … … … $t \geq 0$, so only show Quadrant 1 and (if needed) Quadrant 4

… … … … Position on the vertical axis

0608posntimegrph.gif

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h) .. Draw a Velocity-Time Graph

… … … A Velocity-Time Graph always has:

… … … … Time on the horizontal axis.

… … … … $t \geq 0$, so only show Quadrant 1 and (if needed) Quadrant 4

… … … … Velocity on the vertical axis

… … … … The Velocity-Time Graph is the Derivative Graph of the Position-Time Graph

0608veltimgrph.gif

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