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Name: Samuel Pius

Candidate no: 0038100013

IB Mathematics HL
Internal Assessment

(Freight Elevator)

By (Samuel Pius)
IB Student No: 0038100013
May 2015

Name: Samuel Pius

Candidate no: 0038100013

Topic: Analyzing a freight elevator used to transport cars to
their parking lots
...
Therefore,
I analyzed the freight elevator (vehicle elevators) of my society
...
Vehicular elevators are used
majorly in buildings with limited space to move cars in to the parking garage
...
My society’s freight elevator has a capacity of 2
...


Name: Samuel Pius

Candidate no: 0038100013

The three basic properties in terms of time, which I am going to investigate, are displacement,
velocity and acceleration
...

Furthermore, I was intrigued after noticing the relationship between them: the model of
velocity and displacement is the first and second derivation of the equation of displacement
respectively
...
Therefore, I first enquired the total height of the six
storeyed parking lot and then I divided it by 6 to get an estimated height of each floor
...

Parameter:
Domain: 0 ≤ t ≤ 30; Range: 0 ≤ s ≤ 25
Total height of the parking lot: 21
...

Height per floor: 3
...

Time(seconds)

Displacement(m)
0

0

1

1
...
08

3

3
...
88

5

6
...
30

7

8
...
76

9

10
...
20

11

13
...
60

13

15
...
08

15

18
...
52

17

20
...
65

Name: Samuel Pius

Candidate no: 0038100013
19

21
...
35

21

20
...
91

23

16
...
87

25

10
...
93

27

5
...
05

29

1
...
Then I found the equation of the curve
using the Microsoft office application Excel
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0001x5 + 0
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0219x3 + 0
...
1335x
Displacement/m

20

15

10

5

0
0

5

10

15

20

25

30

Time/s

Hence, I recorded the time taken for the lift to raise in every 3 seconds to reduce the root
mean square error
...
0012x4 - 0
...
0409x2 + 1
...

Finding velocity and acceleration:
In order to find both the velocity (v) and the acceleration (a), it is important to know the
definition of these terms to understand them:
Displacement (s) is the distance moved in a particular direction, which in this model is a
measure of meters downwards
...
To put it into perspective, I will derive
the equation of velocity and acceleration using the equation of displacement,
s= 1E-06x6 - 7E-05x5 + 0
...
0044x3 - 0
...
4143x
Firstly for velocity (v):
v=>

𝑑(𝑠)
𝑑𝑡

=
=

𝑑(1𝐸−06𝑥 6 − 7𝐸−05𝑥 5 + 0
...
0044𝑥 3 − 0
...
4143𝑥)
𝑑𝑡
𝑑(6𝐸−06𝑥 5 )
𝑑𝑡

−

𝑑(3
...
8𝐸−03𝑥 3 )
𝑑𝑡

−

𝑑(0
...
0818𝑥 1 )
𝑑𝑡

+

𝑑(1
...
5𝐸 − 04𝑥 4 + 4
...
0132𝑥 2 − 0
...
4143

Name: Samuel Pius

Candidate no: 0038100013

Therefore, by carrying out the first derivative test on the equation of displacement I derived
the general equation for velocity
...
5𝐸−04𝑥 4 )

𝑑(1
...
8𝐸−03𝑥 3 )

𝑑(0
...
0264𝑥)
𝑑𝑡

𝑑(0
...
0818𝑥 1 )
𝑑𝑡

+

𝑑(1
...
0818)
𝑑𝑡

a= 3𝐸 − 05𝑥 4 − 1
...
0144𝑥 2 − 0
...
0818
Now we have the following expressions:
Displacement(s):
s= 1E-06x6 - 7E-05x5 + 0
...
0044x3 - 0
...
4143x
Velocity (v):
v= 6𝐸 − 06𝑥 5 − 3
...
8𝐸 − 03𝑥 3 − 0
...
0818𝑥1 + 1
...
4𝐸 − 03𝑥 3 + 0
...
0264𝑥 − 0
...

It is important to note at the beginning, that all the variables are vectors, which means that not
only the value but also a sign (direction is important)
...

Inference from the graph labelled figure 1:


The starting point of the elevator is at zero displacement, for velocity at 1
...
082
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

There is no acceleration at t=5
...
Velocity has stopped decreasing because it
reached its stationary point and the displacement is 6
...




Acceleration is negative after t=5
...
551, after that acceleration again starts to increase since it is positive again
...




When it was 15
...
This means that the
elevator had stopped moving at the top of the shaft where the displacement was
maximum
...
580 seconds the elevator starts to descend where in the velocity starts to
increase since it is negative, the acceleration is still decreasing and the magnitude of
the displacement will decrease
...




Explaining the difference between the zero, negative and positive values in the
velocity graph:
When the value of velocity is negative: it means that the velocity of the elevator is
increasing because it is moving downwards
...

When the value of the velocity is zero: it means that the elevator has stopped moving
because it is either at the topmost level or at the ground level
...
When the elevator is at rest:
This situation occurs when the elevator reaches the bottom or the top
...
Initial
situation when velocity equals zero and acceleration at point t=0 does not play
any role
...
When the elevator is speeding:
The velocity and acceleration had different signs expect for the time when the
elevator was descending
...

3
...


Conclusion and Evaluation of this model:
The main problem of a model is that it takes more time to move upwards than to go
downwards, which is ineffective, since over a period this will damage the elevator
...
There are many
constraints in this model hence I will try to improve it by modeling a freight elevator
...



Propertied which should be ignored:
a) The gravitational force (it will not affect the velocity, acceleration or displacement
of the elevator
...




Quantities which will be redesigned:
a) The velocity at which it approaches the ground level must be zero
...

c) Motion of the elevator must look symmetrical for the displacement-time graph
...
The general
formula of the cosine for this is, 𝑠 = 𝐴𝑐𝑜𝑠𝐵(𝑡 + 𝐶) + 𝐷
...
It is the maximum displacement from the principle axis
...

The graph reflects when A<0
...

C= It horizontally translates the graph
...

When the C is negative, the graph translates horizontally to the left
...

When the D is positive, the graph translates vertically upwards
...

Amplitude is the distance between the principle axis and the maximum point hence, our
amplitude is 7
...
Then we add +7
...
Therefore, we get the function as s=7
...
79, which gives us the
following graph:

The graph now that shows the topmost and the ground levels
...
In order to that, we need to alter the period
...
Here B is the desired period that is 10 in this case
...
628
...
79×cos (0
...
79
...
This gives us the
following graph:

Name: Samuel Pius

Candidate no: 0038100013

In order to find the equation for velocity and acceleration we need to derivate the following
equation twice respectively: s= 7
...
628t) +7
...

Firstly for velocity (v):
v=>

𝑑(𝑠)
𝑑𝑡

=
=

v=

𝑑(7
...
628𝑡) +7
...
79×cos (0
...
79×sin (0
...
79)
𝑑𝑡

𝑑(0
...
89×sin (0
...

For Acceleration (a):
a=>

a=

𝑑(𝑣)
𝑑𝑡

=

𝑑(−4
...
628𝑡))
𝑑𝑡

𝑑(−4
...
628𝑡)
𝑑𝑡

×

𝑑(0
...
07 × cos (0
...

Now we have the following expressions:
Displacement(s):

Name: Samuel Pius

Candidate no: 0038100013

s= 7
...
628t) +7
...
89 × sin (0
...
07 × cos (0
...


Acceleration-Time graph:
Domain: 0 ≤ x ≤ 30 and Range: -4 ≤ y ≤ 4
...


Key:
The blue color is the displacement- time graph;
the orange color is the velocity- time graph;
the black color is the acceleration- time graph;

Hence, using the cosine function we simulated the model of a freight elevator used to
transport car to its parking lot
...

a) The velocity at which it approaches the ground level is at zero
...

c) Motion of the elevator look symmetrical for the displacement-time graph
...


Name: Samuel Pius

Candidate no: 0038100013

Application of the model:
This model can be systematically analyzed which could then be modified to make another lift
that is a stair case lift for the material handling system
...
1

1

http://www
...
com/upload/july/7_MODELING
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allaboutcircuits
...
16098/
http://www
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no/files/200810_math_ia_Type_2
...
itu
...
pdf



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