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Page 1
UTA010 (ED
-
II)
Dynamics for the Catapult(No Drag)
Sandeep K SharmaMED, Thapar University, Patiala
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Page 2
Modelling
For a given size, can we maximise thedistance?
What are the key parameters thatcontrol the distance?
Can we formulate a model that will helpus design our Catapult?
Sandeep K SharmaMED, Thapar University, Patiala
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Page 3
Objective of Modelling
create the simplest possible projectilemotion model using standard kinematicformulas and variables.
-initial height
- initial speed
-
initial angle- time step
Sandeep K SharmaMED, Thapar University, Patiala
Animation of themodel for ease ofunderstanding
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Page 4
Fundamentals
force = mass x acceleration (ma)
work = force x distance (Fs)
energy== workpower = rate of work (work/time)
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Page 5
Derived Units
Force (1N=1kgm/s2)
Work (1J=1Nm=1kgm2/s2)
Energy (J)Power (1W=1J/s)
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Page 6
Dynamics
Sandeep K SharmaMED, Thapar University, Patiala
Uniformly accelerated body moving along axis X
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Dynamics
Speed av=distance/time
Accelerationav=velocity/time
Sandeep K SharmaMED, Thapar University, Patiala
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Dynamics
Can derive equations for linear motion(for constant acceleration)
v = u + at
s = ut + 1/2at2
v2 = u2 + 2as
u=initial velocity
v=final velocity
t=time duration
a=acceleration
s=distance travelled
Sandeep K SharmaMED, Thapar University, Patiala
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Dynamics
Example 1: (1-D)
Kick a ball straight up. Given a giveninitial velocity, how high will it go?
DISTANCE.. ????
Sandeep K SharmaMED, Thapar University, Patiala
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Dynamics
Example 1: (1-D)
Use equation:v2=u2+2as
s=u2/2g
a=-g
u
v=0 (at top)
s=?
Sandeep K SharmaMED, Thapar University, Patiala
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Dynamics
Example 2: (1-D)
Drop a rock from a cliff of
height s. How long will ittake to hit the ground/sea?
Time.???
s = ut + 1/2at2
Sandeep K SharmaMED, Thapar University, Patiala
s=1/2at2
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Dynamics
Example 2: (1-D)
Reverse Case
Use equation:
s=ut+1/2at2
s=1/2at2
t (from stopwatch)
u=0 (at top)
s=?
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Dynamics
Example 2: (1-D)
s=1/2at2
t (from stopwatch)
u=0 (at top)
s=?
Example Result: t=3s =>s=44m
However!
t=2.5s =>s=31m
t=3.5s =>s=60m
Sensitive to error: proportional tosquare of t!
Sandeep K SharmaMED, Thapar University, Patiala
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Dynamics
Sandeep K SharmaMED, Thapar University, Patiala
0
10
20
30
40
50
60
70
80
90
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4
Time
S
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Page 15
Dynamics
Can we use these equations to modelthe trajectory of the missile?
And hence predict the distance?
A 2-D problem!
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Page 16
Dynamics (2-D)
y
x
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Page 17
Dynamics (2-D)
y
x
Discretise the curve
1
2
3 4
s
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Page 18
Dynamics (2-D)
y
x
Not u and v now but
v1, v2, v3, v4, etc..
1
2
3 4
v1
v2
v3v4
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Page 19
Dynamics (2-D)
y
x
We can decompose vectors (v, s, a)into x, y components
1
2
3 4
s1x
s1s1y
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Page 20
Dynamics (2-D)
v=u+at becomes:
vx2=vx1+ax1t
vy2=vy1+ay1t
s=ut+1/2at2 becomes:
sx=vx1t+1/2ax1t2
sy=vy1t+1/2ay1t2
Acceleration is constant
(for no drag of lift wellreturn to this point later)
ax=0!
ay=-g
t2-t1= t(keep time interval constantthroughout the flight)
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Page 21
Dynamics (2-D)
s2=s1+vavt
s2=s1+ (v1+(at)/2)t
s2=s1+v1t +(at2)/2
s=s2-s1s= v1 t +(at2)/2
x2=x1+v1xt +(axt2)/2
y2=y1+v1yt +(ayt2
)/2
Sandeep K SharmaMED, Thapar University, Patiala
For constant acceleration-
a=(v2-v1)/(t2-t1)
v2=v1+at
s2=s1+vavt
where vav=(v1+v2)/2
vav=(v1+v1+at)/2
vav=v1 +(at)/2
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Page 22
Dynamics (2-D)
s2=s1+vavt
s2=s1+ (v1+(at)/2)t
s2=s1+v1t +(at2)/2
s=s2-s1s= v1 t +(at2)/2
x2=x1+v1xt +(axt2)/2
y2=y1+v1yt +(ayt2
)/2
Sandeep K SharmaMED, Thapar University, Patiala
sx=vx1t+1/2ax1t2
sy=vy1t+1/2ay1t2
For constant acceleration-
a=(v2-v1)/(t2-t1)
v2=v1+at
s2=s1+vavt
where vav=(v1+v2)/2
vav=(v1+v1+at)/2
vav=v1 +(at)/2
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Page 23
Dynamics Assignment1
Use Excel to study trajectory of missile
Position t x y vx vy ax ay
Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81
delt t 0.01
theta (degrees) 30.00
theta (radians) 0.52
Input Data
Initial Conditions
vx=Vel*cos(theta)
vy=Vel*sin(theta)
Position t x y vx vy ax ay
Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81
delt t 0.01
theta (degrees) 30.00
theta (radians) 0.52
Input Data
Initial Conditions
vx=Vel*cos(theta)
vy=Vel*sin(theta)
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Page 24
Dynamics
t2=t1+t
Position t x y vx vy ax ay
Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81
theta (degrees) 30.00
theta (radians) 0.52
Input Data
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Page 25
Dynamics
x2=x1+vx1t+1/2ax1t2
Position t x y vx vy ax ay
Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81
theta (degrees) 30.00
theta (radians) 0.52
Input Data
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Page 26
Dynamics
y2=y1+vy1t+1/2ay1t2
Position t x y vx vy ax ay
Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81
theta (degrees) 30.00
theta (radians) 0.52
Input Data
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Page 27
Dynamics
vx2=vx1+ax1t
Position t x y vx vy ax ay
Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81
theta (degrees) 30.00
theta (radians) 0.52
Input Data
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Page 28
Dynamics
vy2=vy1+ay1t
Position t x y vx vy ax ay
Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81
theta (degrees) 30.00
theta (radians) 0.52
Input Data
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Dynamics
Const=0!
Position t x y vx vy ax ay
Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81
theta (degrees) 30.00
theta (radians) 0.52
Input Data
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Page 30
Dynamics
Const=-g
Position t x y vx vy ax ay
Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81
theta (degrees) 30.00
theta (radians) 0.52
Input Data
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Page 31
Dynamics
Copy formuladown
Position t x y vx vy ax ay
Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81
delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81
theta (degrees) 30.00 3.00 0.02 0.17 0.10 8.66 4.80 0.00 -9.81
theta (radians) 0.52 4.00 0.03 0.26 0.15 8.66 4.70 0.00 -9.81
5.00 0.04 0.35 0.19 8.66 4.61 0.00 -9.81
6.00 0.05 0.43 0.24 8.66 4.51 0.00 -9.81
7.00 0.06 0.52 0.28 8.66 4.41 0.00 -9.81
8.00 0.07 0.61 0.33 8.66 4.31 0.00 -9.819.00 0.08 0.69 0.37 8.66 4.21 0.00 -9.81
10.00 0.09 0.78 0.41 8.66 4.11 0.00 -9.81
11.00 0.10 0.87 0.45 8.66 4.02 0.00 -9.81
12.00 0.11 0.95 0.49 8.66 3.92 0.00 -9.81
13.00 0.12 1.04 0.53 8.66 3.82 0.00 -9.81
14.00 0.13 1.13 0.57 8.66 3.72 0.00 -9.81
15.00 0.14 1.21 0.60 8.66 3.62 0.00 -9.81
16.00 0.15 1.30 0.64 8.66 3.53 0.00 -9.81
17.00 0.16 1.39 0.67 8.66 3.43 0.00 -9.8118.00 0.17 1.47 0.71 8.66 3.33 0.00 -9.81
19.00 0.18 1.56 0.74 8.66 3.23 0.00 -9.81
Input Data
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Dynamics
Plot x versus yusing chartwizard
Position t x y vx vy ax ay
Vel 10.00 1.00 0.00 0.00 0.00 8.66 5.00 0.00 -9.81
delt t 0.01 2.00 0.01 0.09 0.05 8.66 4.90 0.00 -9.81
theta (degrees) 30.00 3.00 30.01 0.17 0.10 8.66 4.80 0.00 -9.81
theta (radians) 0.52 4.00 30.53 0.26 0.15 8.66 4.70 0.00 -9.81
5.00 30.53 0.35 0.19 8.66 4.61 0.00 -9.81
6.00 30.53 0.43 0.24 8.66 4.51 0.00 -9.81
7.00 30.53 0.52 0.28 8.66 4.41 0.00 -9.81
8.00 30.53 0.61 0.33 8.66 4.31 0.00 -9.81
9.00 30.53 0.69 0.37 8.66 4.21 0.00 -9.81
10.00 30.53 0.78 0.41 8.66 4.11 0.00 -9.81
11.00 30.53 0.87 0.45 8.66 4.02 0.00 -9.81
12.00 30.53 0.95 0.49 8.66 3.92 0.00 -9.81
13.00 30.53 1.04 0.53 8.66 3.82 0.00 -9.81
14.00 30.53 1.13 0.57 8.66 3.72 0.00 -9.81
15.00 30.53 1.21 0.60 8.66 3.62 0.00 -9.81
16.00 30.53 1.30 0.64 8.66 3.53 0.00 -9.81
17.00 30.53 1.39 0.67 8.66 3.43 0.00 -9.8118.00 30.53 1.47 0.71 8.66 3.33 0.00 -9.81
19.00 30.53 1.56 0.74 8.66 3.23 0.00 -9.81
20.00 30.53 1.65 0.77 8.66 3.13 0.00 -9.81
21.00 30.53 1.73 0.80 8.66 3.04 0.00 -9.81
22.00 30.53 1.82 0.83 8.66 2.94 0.00 -9.81
Input Data
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
0.00 2.00 4.00 6.00 8.00 10.00
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Assignment 1
Catapult Dynamics Design Tool using Excel
Individual tutorial exercise
1.Create excel spreadsheet as demonstrated
2.Plot x versus y
3.Study effect of changing velocity
4.Study effect of changing angle
An assignment will be set based on this work. Assignment to be submittedindividually no copying!
Sandeep K Sharmah i i i l