Grafana in space: Monitoring Japan's SLIM moon lander in real time
Ch#4 MOTION IN 2 DIMENSIONS
1. MOTION
IN TWO DIMENSIONS
(PROJECTILE MOTION)
Chapter04 from HRK.
Course code: 301
Course Title: Mechanics and properties of Matter
Course Incharge: SEHRISH INAM
Date: March08,2022
2. PROJECTILE MOTION
What is the path of a projectile as it moves
through the air?
Parabolic?
Straight up and down?
Yes, both are possible.
What forces act on projectiles?
Only gravity, which acts only in the negative
y-direction.
Air resistance is ignored in projectile motion.
3. MOTION IN TWO DIMENSIONS
Using + or – signs is not always sufficient
to fully describe motion in more than one
dimension
Vectors can be used to more fully describe
motion
Still interested in displacement, velocity,
and acceleration
Will serve as the basis of multiple types of
motion in future chapters
4. GENERAL MOTION IDEAS
In two- or three-dimensional
kinematics, everything is the same as
as in one-dimensional motion except
that we must now use full vector
notation
Positive and negative signs are no
longer sufficient to determine the
direction
5. PROJECTILE MOTION
An object may move in both the x and
y directions simultaneously
The form of two-dimensional motion
we will deal with is called projectile
motion
The object which is being moved in 2
dimension simultaneously is known
as projectile.
6. ASSUMPTIONS OF PROJECTILE
MOTION
The free-fall acceleration g is
constant over the range of motion
And is directed downward
The effect of air friction is negligible
With these assumptions, an object in
projectile motion will follow a
parabolic path
This path is called the trajectory
8. VERIFYING THE PARABOLIC
TRAJECTORY
Reference frame chosen
y is vertical with upward positive
Acceleration components
ay = -g and ax = 0
Initial velocity components
vxi = vi cosФ and vyi = vi sinФ
9. VERIFYING THE PARABOLIC TRAJECTORY,
CONT….
Displacements
xf = vxi t = (vi cos Ф) t
yf = vyi t + 1/2ay t2 = (vi sin Ф)t - 1/2 gt2
Combining the equations gives:
This is in the form of y = ax – bx2 which is the
standard form of a parabola
10. ANALYZING PROJECTILE MOTION
Consider the motion as the superposition
of the motions in the x- and y-directions
The x-direction has constant velocity
ax = 0
The y-direction is free fall
ay = -g
The actual position at any time is given
by: rf = ri + vit + 1/2gt2
11. PROJECTILE MOTION VECTORS
rf = ri + vi t + 1/2 g t2
The final position is the
vector sum of the initial
position, the position
resulting from the initial
velocity and the position
resulting from the
acceleration
13. PROJECTILE MOTION IMPLICATIONS
The y-component of the velocity is
zero at the maximum height of the
trajectory
The acceleration stays the same
throughout the trajectory
---------
------------------
14. RANGE &MAXIMUM HEIGHT OF A
PROJECTILE
When analyzing
projectile motion, two
characteristics are of
special interest
The range, R, is the
horizontal distance of
the projectile
The maximum height
the projectile reaches is
h.
15. HEIGHT OF A PROJECTILE,
EQUATION
The maximum height of the projectile
can be found in terms of the initial
velocity.
This equation is valid only for
symmetric motion
16. RANGE OF A PROJECTILE, EQUATION
The range of a projectile can be
expressed in terms of the initial
velocity vector:
This is valid only for symmetric
trajectory
18. RANGE OF A PROJECTILE, FINAL
The maximum range occurs at Ф =
45o
Complementary angles will produce
the same range
The maximum height will be different
for the two angles
The times of the flight will be
different for the two angles.
19. PROJECTILE MOTION – PROBLEM
SOLVING HINTS
Select a coordinate system
Resolve the initial velocity into x and y
components
Analyze the horizontal motion using
constant velocity techniques
Analyze the vertical motion using constant
acceleration techniques
Remember that both directions share the
same time.
20. NON-SYMMETRIC PROJECTILE MOTION
Follow the general rules
for projectile motion
Break the y-direction into
parts
up and down or
symmetrical back to initial
height and then the rest of
the height
May be non-symmetric in
other ways
21. ACCELERATION IN DIFFERENT
FRAMES OF REFERENCE
The derivative of the velocity equation will
give the acceleration equation
The acceleration of the particle measured
by an observer in one frame of reference is
the same as that measured by any other
observer moving at a constant velocity
relative to the first frame.
22. PROJECTILE MOTION
Special case of 2-D motion
Horizontal motion: ax = 0 so vx = constant
Vertical motion: ay = g = constant so the
constant acceleration equations apply.
Assumptions:
Horizontal and vertical motions are
independent of each other
Air resistance (i.e., drag) can be
ignored.
22
25. FREE-FALL ACCELERATION
EQUATIONS
If +y is vertically up, then the free-fall
acceleration due to gravity near Earth’s
surface is a = − g = − 9.8 m/s2.
v = vo − gt
y − yo = vot − ½ gt2
v2 = vo
2 − 2g(y − yo)
y − yo = ½ (vo + v)t
y − yo = vt + ½ gt2 25
26. THE TRAJECTORY OF A PROJECTILE
•What does the free-body diagram look like for force?
Fg
27. THE VECTORS OF PROJECTILE MOTION
What vectors exist in projectile
motion?
Velocity in both the x and y directions.
Acceleration in the y direction only.
vy (Increasing)
vx (constant)
ay = -9.8m/s2
ax = 0
Why is the velocity constant
in the x-direction?
•No force acting on it.
Why does the velocity
increase in the y-direction?
•Gravity.
28. EX. 1: LAUNCHING A PROJECTILE
HORIZONTALLY
A cannonball is shot horizontally
off a cliff with an initial velocity
of 30 m/s. If the height of the
cliff is 50 m:
How far from the base of the cliff
does the cannonball hit the ground?
With what speed does the
cannonball hit the ground?
30. STATE THE KNOWN & UNKNOWN
Known:
vix = 30 m/s
viy = 0 m/s
a = -g = -9.81m/s2
dy = -50 m
Unknown:
dx at y = -50 m
vf = ?
31. PERFORM CALCULATIONS (Y)
Strategy:
Use reference table to find formulas you can use.
vfy = viy + gt
dy = viyt + ½ gt2
Note that g has been substituted for a and y for d.
Use known factors such as in this case where the
initial velocity in the y-direction is known to be zero
to simplify the formulas.
vfy = viy + gt vfy = gt (1)
dy = viyt + ½ gt2 dy = ½ gt2 (2)
32. PERFORM CALCULATIONS (Y)
CONT…
(Use the second formula (2) first
because only time is unknown)
g
d
t
y
2
s
s
m
m
t 2
.
3
)
/
81
.
9
(
)
50
)(
2
(
2
33. PERFORM CALCULATIONS (Y)
CONT…..
Now that we have time, we can use
the first formula (1) to find the final
velocity.
vfy = gt
vy = (-9.8 m/s2)(3.2 s) = -31 m/s
34. PERFORM CALCULATIONS (X)
Strategy:
Since you know the time for the vertical(y-
direction), you also have it for the x-direction.
Time is the only variable that can transition
between motion in both the x and y directions.
Since we ignore air resistance and gravity does not
act in the horizontal (x-direction), a = 0.
Choose a formula from your reference table
dx = vixt + ½ at2
Since a = 0, the formula reduces to x = vixt
dx = (30 m/s)(3.2 s) = 96 m from the base.
35. FINDING THE FINAL VELOCITY (VF)
We were given the initial x-component of
velocity, and we calculated the y-component at
the moment of impact.
Logic: Since there is no acceleration in the
horizontal direction, then vix = vfx.
We will use the Pythagorean Theorem.
vfx = 30m/s
vfy = -31m/s s
m
v
s
m
s
m
v
v
v
v
f
f
fy
fx
f
/
43
)
/
31
(
)
/
30
( 2
2
2
2
vf = ?
36. EX. 2: PROJECTILE MOTION ABOVE THE
HORIZONTAL
A ball is thrown from the top of the Science Wing with
a velocity of 15 m/s at an angle of 50 degrees above
the horizontal.
What are the x and y components of the initial velocity?
What is the ball’s maximum height?
If the height of the Science wing is 12 m, where will the ball
land?
38. STATE THE KNOWN & UNKNOWN
Known:
dyi = 12 m
vi = 15 m/s
= 50°
ay = g = -9.8m/s2
Unknown:
dy(max) = ?
t = ?
dx = ?
viy = ?
vix = ?
39. PERFORM THE CALCULATIONS (YMAX)
y-direction:
Initial velocity: viy = visin
viy = (15 m/s)(sin 50°)
viy = 11.5 m/s
Time when vfy = 0 m/s: vfy = viy + gt (ball at peak)
t = viy / g
t = (-11.5 m/s)/(-9.81 m/s2)
t = 1.17 s
Determine the maximum height: dy(max) = yi + viyt + ½ gt2
dy(max) = 12 m + (11.5 m/s)(1.17 s) + ½ (-9.81 m/s2)(1.17
s)2
dy(max) = 18.7 m
vi = 15 m/s
vxi
vyi
= 50°
40. PERFORM THE CALCULATIONS (T)
Since the ball will accelerate due to gravity over
the distance it is falling back to the ground, the
time for this segment can be determined as
follows
Time from peak to when ball hits the ground:
From reference table: dy(max) = viyt + ½ gt2
Since yi can be set to zero as can viy,
t = 2* dy(max)/g
t = 2(-18.7 m)/(-9.81 m/s2)
t = 1.95 s
By adding the time it takes the ball to reach its
maximum height (peak) to the time it takes to reach
the ground will give you the total time.
ttotal = 1.17 s + 1.95 s = 3.12 s
41. PERFORM THE CALCULATIONS (X)
x-direction:
Initial velocity: vix = vicos
vix = (15 m/s)(cos 50°)
vix = 9.64 m/s
Determine the total distance: x = vixt
dx = (9.64 m/s)(3.12 s)
dx = 30.1 m
vi = 15 m/s
vxi
vyi
= 50°
42. ANALYZING MOTION IN THE X AND Y
DIRECTIONS INDEPENDENTLY.
x-direction y-direction
dx = vix t = vfxt dy = ½ (vi + vf) t
dy = vavg t
vix = vicos vf = viy + gt
dy = viyt + ½g(t)2
vfy
2 = viy
2 + 2gdy
viy = visin