Mathematics Practical Assignment 3

#Mathematics-Practical-Assignment-3
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Part A

#Part-A

Task Outline

#Task-Outline
  • From the collection of vases, select 3 to find the volume of
  • Take measurments of the circumference at approximate points points along the vase. It could be that you consider the vase in multiple sections. Calculate the radius from the circumference.
  • Enter the distance along the vase where the circumference was measured and the matching radius to an external program/ calculator to estimate the function. Again this is best done in relevent sections.
  • Using the functions, calculate the volume of the vase.

Procedual Overview

#Procedual-Overview
  • Data Collection
  • Visualize data
  • Segment Data
  • Derive equations for segments (function estimation)
  • Calculate volume around x-axis of segment (using estimated function)
  • Calculate final volume of vase

Data Collection

#Data-Collection

Data was collected using calipers to measure the diameter of the vase a 1 cm intervals (Distance). Multipul trials were conducted (Trial_n) and the average of these trials was calculated (Average_dia). From the averaged diameter the radius was calulated (radius) by @@0@@.

The resulted volume is the total volume of the vase including the vase and volume it holds.

Python Functions

#Python-Functions

These functions are used to assist in processing the data and rendering it for display

# Vase 1 ## Data

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Sections

#Sections

To calculate the volume of the vase we need a function that is as close as possible to the data. This vase will be broken into the following 4 sections (where @@0@@ is the distance in cm).
Section 1: @@1@@
Section 2: @@2@@

Section 1

#Section-1

Region: @@3@@

Estimating Function

#Estimating-Function

Numpy is used to estimate a function using the least squares method

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Calculating volume of first segment

#Calculating-volume-of-first-segment

@@0@@

@@1@@

@@2@@

@@3@@

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Section 2

#Section-2

Region: @@0@@

Estimating Function

#Estimating-Function

Numpy is used to calculate a polynomial trendline with a degree of 2. That function is then graphed over segment data for comparison.

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Calculating volume of second segment

#Calculating-volume-of-second-segment

@@0@@

@@1@@

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Calculating total volume of vase

#Calculating-total-volume-of-vase

@@0@@

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Summary for Vase 1

#Summary-for-Vase-1

The functions @@0@@ and @@1@@.

By applying the trapezoid rule to the raw data a volume of @@2@@ was found.

There was a difference of @@3@@ between the derived functions and trapezoid calculations.

# Vase 2 ## Data

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Sections

#Sections

To calculate the volume of the vase we need a function that is as close as possible to the data. This vase will be broken into the following 4 sections (where @@0@@ is the distance in cm.
Section 1: @@1@@
Section 2: @@2@@
Section 3: @@3@@
Section 4: @@4@@

Section 1

#Section-1

Region: @@0@@

Estimating function

#Estimating-function

Numpy is used to fit a function using a least squares polynomial fit with an order of 3

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Calculating volume of vase 2 section 1

#Calculating-volume-of-vase-2-section-1

@@0@@

@@1@@

@@2@@

@@3@@

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Section 2

#Section-2

Region: @@0@@

Estimating function

#Estimating-function

Numpy is used to fit a function using the least squares method

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Calculating volume of vase 2 section 2

#Calculating-volume-of-vase-2-section-2

@@0@@

@@1@@

@@2@@

@@3@@

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Section 3

#Section-3

Region: @@0@@

Estimating function

#Estimating-function

Numpy is used to fit a function using a least squares polynomial fit with an order of 3

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Calculating volume of vase 2 section 3

#Calculating-volume-of-vase-2-section-3

@@0@@

@@1@@

@@2@@

@@3@@

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Section 4

#Section-4

Region: @@0@@

Estimating function

#Estimating-function

Since the data for this section is horizontal the function can be easily derived as @@1@@

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Calculating volume of vase 2 section 4

#Calculating-volume-of-vase-2-section-4

@@0@@

@@1@@

@@2@@

@@3@@

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Calculating total volume of vase

#Calculating-total-volume-of-vase

@@0@@

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Summary of Vase 2

#Summary-of-Vase-2

The functions @@0@@.

By applying the trapezoid rule to the raw data a volume of @@1@@ was found.

There was a difference of @@2@@ between the functions and trapezoid calculations.

# Vase 3 ## Data

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Calculating volume

#Calculating-volume

This vase fits an polynomial trendline to the third order without the need to segment.

Estimating function

#Estimating-function

Numpy is used to calculate a polynomial trendline with an order of 3. That function is the graphed over the orignal data for comparison.

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Calculating volume of vase 3

#Calculating-volume-of-vase-3

@@0@@

@@1@@

@@2@@

@@3@@

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Summary of Vase 3

#Summary-of-Vase-3

The function @@0@@.

By applying the trapezoid rule to the raw data a volume of @@1@@ was found.

There was a difference of @@2@@ between the functions and trapezoid calculations.

# Part B ## Task Outline 1. Read through notes on the attached (to cover sheet) "Behind the wheel" including the data tables and the newspaper articles. 2. By considering the deccelerations achived by breaking, describe the outcome if a driver travelling at 40 Km/h and a reaction time of 0.75 seconds sees a child run from a driveway 20m ahead. Repeat for speeds of 50 Km/h and 60 Km/h. Please show all calculations, including any formulas derived. 3. Consider the problem posed about Tailgating. If the two second rule was observed, would the colllision have occured?

Outcomes for deceleration acheived by breaking

#Outcomes-for-deceleration-acheived-by-breaking

A driver is traveling down a road and sees a child run from a driveway 20 meters ahead. Describe the outcome if the car was traveling at @@0@@, @@1@@ and @@2@@ @@3@@ with a reaction time of 0.75 seconds.

Converting speeds

#Converting-speeds

@@4@@

@@5@@

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### Calculating deceleration acheived by breaking To calculate the deceleration acheived by breaking from the above data we The deceleration acheived by breaking can be calculated from the above data using the forumla for deceleration. @@0@@

Deriving formula for calculating stopping distance

#Deriving-formula-for-calculating-stopping-distance

@@0@@

@@1@@

### Distance travelled in reaction time #### At 40 Km/h (11.11 m/s) @@0@@

### Outcomes for Honda Accord (2008) The Honda Accord (2008) has a breaking distance of @@0@@ meters at @@1@@).

@@2@@

Traveling at 40 Km/h (11.11 m/s)

#Traveling-at-40-Km/h-(11.11-m/s)

@@3@@

Traveling at @@4@@ there would not have been a collision.

Traveling at 50 Km/h (13.89 m/s)

#Traveling-at-50-Km/h-(13.89-m/s)

@@5@@

Traveling at @@6@@ there would have been a collision.

Traveling at 60 Km/h (16.67 m/s)

#Traveling-at-60-Km/h-(16.67-m/s)

@@7@@

Traveling at @@8@@ there would have been a collision.

### Outcomes for Falcon Futura (2002) The Falcon Futura (2002) has a breaking distance of @@0@@ meters at @@1@@).

@@2@@

Traveling at 40 Km/h (11.11 m/s)

#Traveling-at-40-Km/h-(11.11-m/s)

@@3@@

Traveling at @@4@@ there would not have been a collision.

Traveling at 50 Km/h (13.89 m/s)

#Traveling-at-50-Km/h-(13.89-m/s)

@@5@@

Traveling at @@6@@ there would have been a collision.

Traveling at 60 Km/h (16.67 m/s)

#Traveling-at-60-Km/h-(16.67-m/s)

@@7@@

Traveling at @@8@@ there would have been a collision.

### Outcomes for Mazda 3 (2004) The Mazda 3 (2004) has a breaking distance of @@0@@ meters at @@1@@).

@@2@@

Traveling at 40 Km/h (11.11 m/s)

#Traveling-at-40-Km/h-(11.11-m/s)

@@3@@

Traveling at @@4@@ there would not have been a collision.

Traveling at 50 Km/h (13.89 m/s)

#Traveling-at-50-Km/h-(13.89-m/s)

@@5@@

Traveling at @@6@@ there would have been a collision.

Traveling at 60 Km/h (16.67 m/s)

#Traveling-at-60-Km/h-(16.67-m/s)

@@7@@

Traveling at @@8@@ there would have been a collision.

### Outcomes for Peugeot 307 (2005) The Peugeot 307 (2005) has a breaking distance of @@0@@ meters at @@1@@).

@@2@@

Traveling at 40 Km/h (11.11 m/s)

#Traveling-at-40-Km/h-(11.11-m/s)

@@3@@

Traveling at @@4@@ there would not have been a collision.

Traveling at 50 Km/h (13.89 m/s)

#Traveling-at-50-Km/h-(13.89-m/s)

@@5@@

Traveling at @@6@@ there would have been a collision.

Traveling at 60 Km/h (16.67 m/s)

#Traveling-at-60-Km/h-(16.67-m/s)

@@7@@

Traveling at @@8@@ there would have been a collision.

## Tailgating Car A is a Mazda 3 (2004) travelling at @@0@@ as good as the Mazda. Both drivers have a reaction time of 0.75 seconds. Problems in road conditions ahead cause car A to halt with the full power of their breaks. Describe the outcome. What would the 2 second rule have achieved?

Distance travelled in reaction time of 0.75 seconds at speed of 90 km/h

#Distance-travelled-in-reaction-time-of-0.75-seconds-at-speed-of-90-km/h

@@1@@

@@2@@

Deceleration of (1979 Ford)

#Deceleration-of-(1979-Ford)

Car B has @@3@@). So: @@4@@

Time of impact

#Time-of-impact

There will be an impact at the time when displacement of Car A is equal to Car B.
@@5@@ relative time of collision (to 2 decimal places)} \ &= 1.51\,seconds \ \end{align} $$

Velocity at time of impact

#Velocity-at-time-of-impact

@@6@@

Impact if 2 second rule was observed

#Impact-if-2-second-rule-was-observed

@@7@@

Summary

#Summary

Cars A and B at travelling at @@8@@. If Car B had followed the 2 second rule a collision would not have occured.