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 particle’s position, displacement,
velocity, and acceleration at any instant in time.

4. Rectilinear Kinematics:
Continuous Motion
Lecture Outline
Revision:
kinematics
Example:
Kinematics
POSITION
• A particle’s 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

5. Rectilinear Kinematics:
Continuous Motion
VELOCITY
Lecture Outline
Revision:
kinematics
•
Example:
Kinematics
Erratic motion:
Graphical
method
Example:
Graphical
method
Projectile
motion
Example:
Projectile
motion
SPEED

6. Rectilinear Kinematics:
Continuous Motion
Lecture Outline
Revision:
kinematics
Example:
Kinematics
Erratic motion:
Graphical
method
Example:
Graphical
method
Projectile
motion
Example:
Projectile
motion
ACCELERATION

7. Rectilinear Kinematics:
Continuous Motion
ACCELERATION
Lecture Outline
Revision:
kinematics
Example:
Kinematics
Erratic motion:
Graphical
method
Example:
Graphical
method
Projectile
motion
Example:
Projectile
motion
+

8. Rectilinear Kinematics:
Continuous Motion
ACCELERATION
Lecture Outline
Revision:
kinematics
Example:
Kinematics
Example from Lecture 3:
Coordinate system
2
vB
2
vA
Erratic motion:
Graphical
method
2aC ( s B
sA )
-9.81 m/s2
Example:
Graphical
method
Projectile
motion
Example:
Projectile
motion
2
vC
2
vB
2aC ( sC
-9.81 m/s2
sB )
+

9. Kinematics
Lecture Outline
by definition:
Revision:
kinematics
Example:
Kinematics
Erratic motion:
Graphical
method
Example:
Graphical
method
Projectile
motion
by rearranging:
Example:
Projectile
motion
9
9

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 can’t 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

25. Erratic motion and graphical methods
Lecture Outline
Revision:
kinematics
Example:
Kinematics
Erratic motion:
Graphical
method
Example:
Graphical
method
Projectile
motion
Example:
Projectile
motion
25

26. Erratic motion and graphical methods
Lecture Outline
Revision:
kinematics
•
When particle’s 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

40. Projectile motion
Lecture Outline
Revision:
kinematics
Example:
Kinematics
Erratic motion:
Graphical
method
Example:
Graphical
method
Projectile
motion
Example:
Projectile
motion
40

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
•
•
•
•
•
Projectile’s 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