1. Faculty of Engineering ENG1040 Engineering Dynamics
Kinematics of a Particle Dr Lau Ee Von Sunway Lecture 4 ENG1040
Engineering Dynamics
2. Lecture outline Lecture Outline Revision: kinematics
Example: Kinematics Erratic motion: Graphical method Example:
Graphical method Projectile motion Example: Projectile motion
Concepts of position, displacement, velocity, and acceleration
Study particle motion along a straight line Erratic motion: the
graphical method Projectile motion two dimensional motion
3. Rectilinear Kinematics: Continuous Motion Lecture Outline
Revision: kinematics Example: Kinematics Erratic motion: Graphical
method Example: Graphical method Projectile motion Example:
Projectile motion RECTILINEAR KINEMATICS Defines a particles
position, displacement, velocity, and acceleration at any instant
in time.
4. Rectilinear Kinematics: Continuous Motion Lecture Outline
Revision: kinematics Example: Kinematics POSITION A particles
position is defined from an origin. We must always define a
coordinate system to a problem. Erratic motion: Graphical method
Example: Graphical method Projectile motion Example: Projectile
motion DISPLACEMENT
10. Kinematics Lecture Outline Revision: kinematics Example:
Kinematics Erratic motion: Graphical method When do we use a dv dt
and a v dv ? ds Example: Find the velocity if s = 2m when t = 0 s
Given a = 20t Given a = 20s Example: Graphical method Projectile
motion Example: Projectile motion 10
11. Kinetics/Kinematics problems... Lecture Outline Revision:
kinematics Example: Kinematics Erratic motion: Graphical method
Example: Graphical method Analysis procedure 1. Establish a
coordinate system 2. Draw Free Body Diagram(s) Graphical
representation of all forces acting on the system. 3. Establish
known & unknown quantities Projectile motion Example:
Projectile motion 4. Apply Equation(s) of Motion in each direction
5. Evaluate kinematics to solve problem
12. Example 12.4 Lecture Outline Revision: kinematics Example:
Kinematics Erratic motion: Graphical method Example: Graphical
method Projectile motion Example: Projectile motion A metallic
particle travels downward through a fluid that extends from plate A
and plate B under the influence of magnetic field. If particle is
released from rest at midpoint C, s = 100 mm, and acceleration, a =
(4s) m/s2, where s in meters, determine velocity when it reaches
plate B and time need to travel from C to B.
13. Example 12.4 Lecture Outline Revision: kinematics Example:
Kinematics Erratic motion: Graphical method Example: Graphical
method Step 1: choose a coordinate system Projectile motion
Example: Projectile motion
14. Example 12.4 Lecture Outline Revision: kinematics a v
Example: Kinematics dv dt ds dt a dv v ds Erratic motion: Graphical
method Example: Graphical method Step 2: employ kinematics
Projectile motion Example: Projectile motion a = (4s) m/s2
15. Example 12.4 Lecture Outline Revision: kinematics a v
Example: Kinematics dv dt ds dt a dv v ds Erratic motion: Graphical
method Example: Graphical method Step 2: employ kinematics
Projectile motion Example: Projectile motion a = (4s) m/s2 a dv v
ds 4s dv v ds
16. Example 12.4 Lecture Outline Revision: kinematics a
Example: Kinematics Erratic motion: Graphical method Example:
Graphical method Projectile motion Example: Projectile motion dv dt
v ds dt a dv v ds Step 2: employ kinematics a dv v ds s dv v ds 4s
v 4sds sinitial vdv vinitial
17. Example 12.4 Lecture Outline Revision: kinematics a v
Example: Kinematics Erratic motion: Graphical method Example:
Projectile motion ds dt a dv v ds Step 2: employ kinematics
Example: Graphical method Projectile motion dv dt s v 4sds sinitial
2s 2 s sinitial vdv vinitial 1 2 v 2 v vinitial
18. Example 12.4 Lecture Outline Revision: kinematics a v
Example: Kinematics Erratic motion: Graphical method Example:
Graphical method Projectile motion Example: Projectile motion dv dt
ds dt a dv v ds Step 2: employ kinematics s v 4sds sinitial 2s 2 s
0.1 vdv vinitial 1 2 v 2 v 0 Only put in initial limits
19. Example 12.4 Step 2: employ kinematics Lecture Outline
Revision: kinematics Example: Kinematics Erratic motion: Graphical
method 1 2 v 2 v 2s 2 s Example: Graphical method Projectile motion
Example: Projectile motion 2 2 a 2 0.1 0.01 Leave it in the general
form of equation dv dt v ds dt a dv v ds Substitute sb = 200mm =
0.2m to find vb vb 0.346 m / s
20. Example 12.4 Lecture Outline Revision: kinematics Example:
Kinematics Erratic motion: Graphical method Example: Graphical
method Projectile motion Example: Projectile motion A metallic
particle travels downward through a fluid that extends from plate A
and plate B under the influence of magnetic field. If particle is
released from rest at midpoint C, s = 100 mm, and acceleration, a =
(4s) m/s2, where s in meters, determine velocity when it reaches
plate B and time need to travel from C to B.
21. Example 12.4 Time to reach plate B? Lecture Outline
Revision: kinematics a Example: Kinematics Erratic motion:
Graphical method a Example: Graphical method 2s Projectile motion
Example: Projectile motion v dv v ds 2 s 4s 1 2 v 2 2 s 0.1 2 dv v
ds dv dt v ds dt a dv v ds v 0 0.01 Use general form of
equation!
22. Example 12.4 Lecture Outline Time to reach plate B?
Revision: kinematics v Example: Kinematics a dv dt v ds dt a dv v
ds ds v dt Erratic motion: Graphical method 2s Example: Graphical
method s 0.1 Projectile motion Example: Projectile motion 2 s 2
0.01 t 2 0.01 ds s 2 0.1 t 0.5 0 0.5 dt 2 dt ln (0.2) 2 0.01 s 2
Only put in initial limits 2.303 Leave it in the general form of
equation
23. Example 12.4 Lecture Outline Revision: kinematics
Substitute sb = 200mm = 0.2m to find tb Example: Kinematics Erratic
motion: Graphical method Example: Graphical method Projectile
motion Example: Projectile motion t = 0.658s a dv dt v ds dt a dv v
ds NOTE: Why cant we use v u at s s0 ut 1 2 ? at 2 Acceleration is
NOT a constant (a = 4s)
24. Faculty of Engineering ENG1040 Engineering Dynamics Erratic
motion and graphical methods Dr Greg Sheard - Clayton Dr Lau Ee Von
- Sunway Lecture 4 ENG1040 Engineering Dynamics
26. Erratic motion and graphical methods Lecture Outline
Revision: kinematics When particles motion is erratic, it is
described graphically using a series of curves A graph is used to
described the relationship with any 2 of the variables: Example:
Kinematics Erratic motion: Graphical method Example: Graphical
method Projectile motion Example: Projectile motion a, v, s, t a We
use dv dt v ds dt a dv v ds
27. Example 12.6 Lecture Outline Revision: kinematics Example:
Kinematics Erratic motion: Graphical method Example: Graphical
method Projectile motion Example: Projectile motion A bicycle moves
along a straight road such that it position is described by the
graph as shown. Construct the v-t and a-t graphs for 0 t 30s.
28. Example 12.6 Lecture Outline Revision: kinematics Example:
Kinematics Erratic motion: Graphical method Example: Graphical
method Projectile motion Example: Projectile motion A bicycle moves
along a straight road such that it position is described by the
graph as shown. Construct the v-t and a-t graphs for 0 t 30s. a dv
dt ds v dt dv a v ds
29. Example 12.6 Lecture Outline Revision: kinematics Example:
Kinematics Erratic motion: Graphical method Example: Graphical
method Projectile motion Example: Projectile motion Solution v-t
Graph By differentiating the equations defining the s-t graph, we
have ds v 0.6t 2 0 t 10s; s 0.3t dt ds 10s t 30s; s 6t 30 v 6
dt
30. Example 12.6 Lecture Outline Solution Revision: kinematics
Example: Kinematics a-t Graph Erratic motion: Graphical method By
differentiating the eqns defining the lines of the v-t graph,
Example: Graphical method Example: Projectile motion 0.6t a 10
Projectile motion 0 t 10 s; v 6 a t 30 s; v dv dt dv dt 0.6 0 a dv
v ds a dv dt
31. Example 12.7 Lecture Outline Revision: kinematics Example:
Kinematics Erratic motion: Graphical method Example: Graphical
method Projectile motion Example: Projectile motion A test car
starts from rest and travels along a straight track such that it
accelerates at a constant rate for 10 s and then decelerates at a
constant rate. Draw the v-t and s-t graphs and determine the time t
needed to stop the car. How far has the car traveled?
32. Example 12.7 Lecture Outline Revision: kinematics Example:
Kinematics Erratic motion: Graphical method Example: Graphical
method Projectile motion Example: Projectile motion A test car
starts from rest and travels along a straight track such that it
accelerates at a constant rate for 10 s and then decelerates at a
constant rate. Draw the v-t and s-t graphs and determine the time t
needed to stop the car. How far has the car traveled? a dv dt v ds
dt a dv v ds
33. Example 12.7 Lecture Outline Revision: kinematics Example:
Kinematics Erratic motion: Graphical method Example: Graphical
method Projectile motion Solution v-t Graph Using initial condition
v = 0 when t = 0, v 0 t 10s a 10; 0 dv t 0 10 dt , v 10 t When t =
10s, v = 100m/s, 10s t t; a v 2; 100 v dv t 10 2 dt , 2t 120
Example: Projectile motion a dv dt
34. Example 12.7 Solution Lecture Outline Revision: kinematics
Example: Kinematics Erratic motion: Graphical method Example:
Graphical method s-t Graph. Using initial conditions s = 0 when t =
0, s 0 t 10s; v 10t; 0 t ds 0 5t 2 When t = 10s, s = 500m, 10s t
60s; v s 2t 120; ds 500 Projectile motion 10 t dt , s s t 10 2t 120
dt t 2 120 t 600 Example: Projectile motion v ds dt
35. Example 12.7 Lecture Outline Revision: kinematics Example:
Kinematics Erratic motion: Graphical method Example: Graphical
method Projectile motion Example: Projectile motion A test car
starts from rest and travels along a straight track such that it
accelerates at a constant rate for 10 s and then decelerates at a
constant rate. Draw the v-t and s-t graphs and determine the time t
needed to stop the car. How far has the car traveled? a dv dt v ds
dt a dv v ds
36. Example 12.7 Lecture Outline Revision: kinematics Time
needed to stop the car? Solution Example: Kinematics Erratic
motion: Graphical method Example: Graphical method Projectile
motion Example: Projectile motion 10s t t; v 2t 120 When t = t, v =
0 t = 60 s
37. Example 12.7 Lecture Outline Revision: kinematics Example:
Kinematics Erratic motion: Graphical method Example: Graphical
method Projectile motion Example: Projectile motion A test car
starts from rest and travels along a straight track such that it
accelerates at a constant rate for 10 s and then decelerates at a
constant rate. Draw the v-t and s-t graphs and determine the time t
needed to stop the car. How far has the car traveled? a dv dt v ds
dt a dv v ds
38. Example 12.7 Lecture Outline Revision: kinematics Total
distance travelled? Solution Example: Kinematics Erratic motion:
Graphical method Example: Graphical method Projectile motion
Example: Projectile motion 10 s t 60 s; s t 2 120 t 600 When t = t
= 60s, s = 3000m
39. Faculty of Engineering ENG1040 Engineering Dynamics
Projectile motion Dr Greg Sheard Clayton Dr Lau Ee Von Sunway
Lecture 4 ENG1040 Engineering Dynamics
41. Projectile motion Lecture Outline Revision: kinematics
Example: Kinematics We can resolve the velocity or acceleration to
its x and y directions, and vice versa. a dv dt v ds dt a dv v ds
Erratic motion: Graphical method Example: Graphical method
Projectile motion Example: Projectile motion 41
42. Projectile motion Lecture Outline Some simplifications (for
ENG1040) Revision: kinematics Example: Kinematics Erratic motion:
Graphical method Example: Graphical method Projectile motion
Example: Projectile motion Projectiles acceleration always acts
vertically Projectile launched at (x0, y0) and path is defined in
the x-y plane Fluid resistance is neglected Only force is its
weight downwards ac = g = 9.81 m/s2 (constant downwards
acceleration)
43. 12.6 Motion of Projectile Lecture Outline Revision:
kinematics Horizontal Motion Since ax = 0, Example: Kinematics
Erratic motion: Graphical method v Example: Graphical method x
Projectile motion Example: Projectile motion We can use the
constant acceleration equations v2 v0 ac t ; 1 2 x0 v0t ac t ; 2 2
v0 2ac ( s s0 ); vx (v0 ) x x x0 (v0 ) x t vx (v0 ) x Horizontal
component of velocity remain constant during the motion
44. 12.6 Motion of Projectile Lecture Outline Revision:
kinematics Example: Kinematics Erratic motion: Graphical method
Example: Graphical method Projectile motion Example: Projectile
motion Vertical Motion Positive y axis is upward, thus ay = - g
Once again, we can use the constant acceleration equations: v v0 ac
t ; y v 2 y0 2 0 v 1 2 v0t ac t ; 2 2ac ( y y0 ); vy y 2 vy (v0 ) y
gt 1 2 y0 (v0 ) y t gt 2 (v0 ) 2 2 g ( y y0 ) y
45. 12.6 Motion of Projectile Lecture Outline Revision:
kinematics Example: Kinematics Erratic motion: Graphical method
Example: Graphical method Projectile motion Example: Projectile
motion PROCEDURE FOR ANALYSIS 1. Establish a coordinate system 2.
Sketch the trajectory of the particle 3. Specify 3 unknowns and
data between any two points on the path 4. Employ the equations of
motion 5. Acceleration of gravity always acts downwards 6. Express
the particle initial and final velocities in the x, y components
Note: Positive and negative position, velocity and acceleration
components always act in accordance with their associated
coordinate directions
46. Example Lecture Outline Revision: kinematics Example:
Kinematics Erratic motion: Graphical method Example: Graphical
method Projectile motion Example: Projectile motion The chipping
machine is designed to eject wood chips vO = 7.5 m/s. If the tube
is oriented at 30 from the horizontal, determine how high, h, a
chip is when it is 6 metres away (horizontally) from the tube.
47. Example Lecture Outline Revision: kinematics Example:
Kinematics Erratic motion: Graphical method The chipping machine is
designed to eject wood chips at vO = 7.5 m/s. If the tube is
oriented at 30 from the horizontal, determine how high a chip is
when it is 6 metres away (horizontally) from the tube. Step 1:
Establish a coordinate system: Example: Graphical method Projectile
motion Example: Projectile motion y x
48. Example Lecture Outline Revision: kinematics Example:
Kinematics Erratic motion: Graphical method Example: Graphical
method Projectile motion Example: Projectile motion The chipping
machine is designed to eject wood chips at vO = 7.5 m/s. If the
tube is oriented at 30 from the horizontal, determine how high a
chip is when it is 6 metres away (horizontally) from the tube. Step
2: Determine the vertical and horizontal components of initial
velocity (vO ) x (7.5 cos30 ) (vO ) y (7.5 sin 30 ) 3.75m / s 6.5m
/ s
49. Example Lecture Outline Revision: kinematics Example:
Kinematics Erratic motion: Graphical method Example: Graphical
method Projectile motion Example: Projectile motion The chipping
machine is designed to eject wood chips at vO = 7.5 m/s. If the
tube is oriented at 30 from the horizontal, determine how high a
chip is when it is 6 metres away (horizontally) from the tube. Step
3: Apply the (relevant) equations of motion yA yO (v0 ) y tOA 1 2
gtOA 2 1 equation, 2 unknowns Vertical motion vy (v0 ) y y y0 (v0 )
y t 2 vy (v0 ) 2 y gt 1 2 gt 2 2 g ( y y0 ) Remember: these are
just simplified constant acceleration equations
50. Example Lecture Outline Revision: kinematics Example:
Kinematics Erratic motion: Graphical method The chipping machine is
designed to eject wood chips at vO = 7.5 m/s. If the tube is
oriented at 30 from the horizontal, determine how high a chip is
when it is 6 metres away (horizontally) from the tube. Step 3:
Apply the (relevant) equations of motion Example: Graphical method
vx xA Projectile motion Example: Projectile motion Horizontal
motion x0 (v0 ) x tOA tOA 0.923s (v0 ) x x x0 (v0 ) x t vx (v0 ) x
Remember: these are just simplified constant acceleration
equations
51. Example Lecture Outline Revision: kinematics Example:
Kinematics Erratic motion: Graphical method Example: Graphical
method Projectile motion Example: Projectile motion The chipping
machine is designed to eject wood chips at vO = 7.5 m/s. If the
tube is oriented at 30 from the horizontal, determine how high a
chip is when it is 6 metres away (horizontally) from the tube. Step
3: Apply the (relevant) equations of motion tOA yA 0.9231 s yO h
1.38m (v0 ) y tOA Vertical motion vy 1 2 gtOA 2 (v0 ) y y y0 (v0 )
y t 2 vy (v0 ) 2 y gt 1 2 gt 2 2 g ( y y0 ) Remember: these are
just simplified constant acceleration equations
52. Conclusions We have considered the rectilinear equations
for kinematics in three situations: 1 dimensional motion
rectilinear, continuous motion 1 dimensional motion erratic motion
The Graphical method 2 dimensional motion Projectile motion 52
