Optimising Flight Times

flight path

Calculate the best flight time from A to B and reduce greenhouse gas emissions!

The table below represents a cross section through the atmosphere and gives wind speeds (in m/s) in boxes which are 200km long and 1km high.

Your task is to pilot an aircraft, which flies at 230m/s when it is flying in the less dense atmosphere higher than 5km, and 150m/s when it is flying in the more dense atmosphere lower than 5km, from A to B in the shortest time possible.

Remember, flying in the same direction as the wind increases your speed but flying against the wind slows you down.

Map your route on the chart below and then calculate the flight time!

Rules

  1. You take off from the ground at A and land on the ground at B.
  2. You can only climb, or descend, one box per 200km.
  3. Give your final answer in hours and minutes.
flight data

Some students may find the following table useful:

flight time table

Ocean Warming and Kettles – Teachers’ Notes

Resource produced in conjunction with Sustainability Physics.

Students’ worksheet. 

Motivation:

  • The world’s oceans are heating. Their temperature is not rising as fast as that of the land or air, but they are the major store of the excess thermal energy resulting from greenhouse gas emissions
  • According to the abstract of this study https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2012GL051106#support-information-section the top 700m of the oceans have warmed by 0.18°C on average between 1955 and 2010. This resource investigates how big this store of thermal energy is.

Curriculum links

  • GCSE physics heat capacity, power calculations and estimation
  • GCSE maths standard form: the order of magnitude of the numbers is more important for this question than the numerical values

This could be used as a starter exercise: Can students do the whole question at once given only the radius of the Earth, the temperature rise and the specific heat capacity of sea water?

  1. Why does the temperature of the sea rise more slowly than the temperature of the land or air?
    Water has a higher heat capacity (4kJ/kg/°C for sea water) than land (2kJ/kg/°C for rock) or air (around 1kJ/kg/°C). For the same input of thermal energy, the increase in temperature is smaller for the ocean than it is for the land.
  2. Find the area of the Earth’s oceans using the following information: the radius of the Earth is 6371km and the oceans cover about 70% of the Earth’s surface.
    A = 4πR2 = 3.57 x1014 m2 ≈ 3.6 x1014 m2
    3. Find the volume of the top 700m of the oceans. Ignore all the coastal sections of the ocean which are shallower than 700m.
    V = A*height = 2.5 x17 m3
    4. Find the mass of the top 700m of the ocean. Use the density of seawater as ρ = 1025 kg/m3
    Mass = V* ρ = 2.56 x1020 kg ≈ 2.6 x1020 kg
    5. Find the energy required to give this mass a temperature rise of 0.18°C. The specific heat capacity of sea water is 4 kJ/kg/K
    Energy = C*Mass*ΔT = 1.8 x1023 J
    6. Find the average power over the 55 year heating period
    Power = Energy/time = 1.1 x1014 W
    7. How big is that power? Find the power ‘per person’ by dividing the total power by the number of people on Earth today (8 billion people)
    1.3 x104 W
    8. A kettle has a power of 2.5kW. How many kettles would each person on the Earth have to boil to have the same total power?
    1.3 x104 W / 2.5 x103 W = 5.3 ≈ 5
    The warming of the upper ocean between 1955 and 2010 is equivalent to the energy used by every person on Earth boiling 5 kettles continuously for 55 years! This question only considers the upper ocean. The lower ocean is also warming and storing energy.

Ocean Warming and Kettles

Resource produced in conjunction with Sustainability Physics.

Teachers’ Notes

Motivation

The world’s oceans are heating. Their temperature is not rising as fast as that of the land or air, but they are the major store of the excess thermal energy resulting from greenhouse gas emissions

The top 700m of the oceans have warmed by 0.18°C on average between 1955 and 2010.

This resource investigates how big this store of thermal energy is.

  1. Why does the temperature of the sea rise more slowly than the temperature of the land or air?
  2. Find the area of the Earth’s oceans using the following information: the radius of the Earth is 6400km and the oceans cover about 70% of the Earth.
  3. Find the volume of the top 700m of the oceans. Ignore all the coastal sections of the ocean which are shallower than 700m.
  4. Find the mass of the top 700m of the ocean. Use the density of seawater as ρ = 1025 kg/m3
  5. Find the energy required to give this mass a temperature rise of 0.18°C. The specific heat capacity of sea water is 4 kJ/kg/K
  6. Find the average power over the 55 year heating period
  7. How big is that power? Find the power ‘per person’ by dividing the total power by the number of people on Earth today (8 billion people)
  8. A kettle has a power of 2.5kW. How many kettles would each person on the Earth have to boil to have the same total power?

KS3 Maths – Egypt’s Road Problem

In this resource linked to COP27 in Egypt, maths students apply pythagoras’ theorem to solving Egypt’s road building conundrum. 

Introduction/Motivation

The 2022 United Nations climate change conference (27th session of the Conference of Parties – COP27) will be held in Sharm El-Sheikh in Egypt, starting on the 7th of November.

In the introduction video screened at the end of COP26 in Glasgow, Egypt celebrates its road-building project. This resource explores efficient road designs and the climate impacts of car travel.

Resources:

PowerPoint

4 City square template

Created with support from MEI 

Section 1: Lesson Introduction

Show the Egypt introduction video from COP26 and show them the pictures of new Egyptian roads.

Road-building clip only:

Or the full Egypt introductory clip: from 09:21-12:26 in https://unfccc-cop26.streamworld.de/webcast/closing-plenary-of-the-cop-followed-by-cmp-and-c-2

Ask students what comments or questions they have on the video: What do they wonder?

They could discuss in pairs or groups before giving feedback to the class.
You could steer the discussion towards some of the following points:

  • What are the advantages and disadvantages of building new roads?
  • What do you thing about building new roads compared to the other climate mitigation and adaptation projects mentioned in the video?
  • Roads for sustainable development: connecting cities and industry
  • Will a new road reduce traffic?
  • Building roads versus building railways/airports
  • How will building new roads impact greenhouse gas emissions?
  • Should houses be demolished to make way for new roads? https://www.reuters.com/world/middle-east/egypts-road-building-drive-eases-jams-leaves-some-unhappy-2021-05-14/
  • How should governments decide which new roads to build? How can we reduce travel time for the most people, reduce the length of the new road or reduce the greenhouse gas emissions from people travelling on the road?

The remainder of the lesson uses maths to explore the last point.

Section 2: Scenario motivation for the Steiner problem

This could be introduced as motivation for the Steiner problem, or as a real world application once the problem has been solved (after section 3).

One of the new roads connects Beni Suef and Zaafarana. https://scoopempire.com/where-to-%EF%BB%BFegypt-launches-a-series-of-road-and-construction-projects-to-link-up-cities-far-and-wide/

Together with the important cities of Cairo and Suez, this can be simplified and framed as an example of the famous ‘Steiner Problem’: 

Source: Google maps

Which looks a bit like:

 

Steiner problem

Section 3: Steiner Problem using a Square


To simplify the problem, start by investigating 4 cities in a square. What is the most efficient way to connect all the cities (using the shortest distance of road)? You need to be able to visit all the cities on the road network, but you can go via other cities.

This problem is also described here: https://nrich.maths.org/14937

Students can use the sheet of squares (or squared paper) to draw as many different designs as they can think of, using curves and straight lines, or just straight lines.

Collect some ideas and ask students to calculate the total road distance required. For the square side length, you could use realistic road-distance numbers (eg 100km), simple numbers (eg 10, 1) or a symbol such as x. Students could first measure the distances using a ruler, then calculate them using Pythagoras’ theorem.

Help students to arrive at the optimal solution by considering the two designs below: Is there an intermediate design that would be even better?

Once students have arrived at the optimal solution, this video gives a good demonstration using soap bubbles: https://www.youtube.com/watch?v=dAyDi1aa40E

Section 4: Context Calculations

At this point you could choose to start using realistic road distances or the fact that the cities are not in a square but are closer to a rectangle (see the first extension point below).

  • What is the total distance of the optimal network? How long would it take to travel between each city whilst travelling at the Egyptian motorway speed limit of 100km/hr? (https://www.autoeurope.ie/driving-information-egypt)
  • Assume that the roads connecting Beni Suef to Cario, Cairo to Suez and Suez to Zaafarana already exist. Which one new road should be built to reduce the travel time from Beni Suef to Zaafarana? What is the reduction in travel time?
  • Cars emit around 120 gCO2/km (https://www.eea.europa.eu/data-and-maps/indicators/average-co2-emissions-from-motor-vehicles/assessment-1)
    By how much does the new road reduce the CO2 emissions of a journey from Beni Suef to Zaafarana?
  • Due to the shorter travel time, the new road might increases the number of journeys between Beni Suef and Zaafarana. How many extra journeys are needed to outweigh the decrease in emissions from the reduced distance?

Extension Ideas

  • Return to the introduction video at the end of the lesson. What do students think about building roads and climate change after completing the activity?
    • The 4 cities in Egypt are not in a perfect square, but are close to being in a rectangle. Does this change the optimal road network? This is discussed at https://thatsmaths.com/2015/01/29/the-steiner-minimal-tree/
  • Students could use google maps to look at the real travel time between the 4 cities using different routes.
  • What happens when we consider more cities?
  • Can students think of other situations where this problem could apply? (gas pipelines, rail networks, broadband cables)
  • Think about the real-world practicalities that the Steiner solution doesn’t address. Is it the best solution if most journeys are between Cairo and Suez? Should existing roads be removed in order to build the most efficient network? Which journey times would be increased by this?

Models for Climate Change

Here is a broad range of simple (ish) climate models suitable for relatively advanced students:

Core Maths – EVolution of vehicle sales

Resource produced in collaboration with MEI

Brief overview of session ‘logic’

  • Explore the infographic – what can be worked out from this information and what questions does it raise?
  • Look at trends in vehicle registrations
  • Look at proportions of types of newly registered vehicles over time – why has the percentage of petrol cars being registered increased from 2015 to 2020?
  • Do some calculations to show that the number of petrol cars being registered has decreased from 2015 to 2020.
  • Reflect upon the implications for misleading representations of data
  • Consider the implications of the ban on new petrol and diesel cars by 2030 – what affect will this ban have on the proportions of car types being registered?
  • What questions does the increase in electric vehicles raise?

Mathematical opportunities offered

  • Interpretation of data, statistics, graphs, infographics in context
  • Critiquing graphs
  • Reading scales
  • Calculating percentages
  • Exploring proportions of quantities over time
  • Making conjectures about future proportions given available data
  • Analysing and comparing data in order to develop and present a conclusion.
Download the resources
  1. Session plan
  2. Presentation
  3. Student sheet

Key Stage 3 – EVolution of vehicle sales

Resource produced in collaboration with MEI

Brief overview of session ‘logic’

  • Explore the infographic – what can be worked out from this information and what questions does it raise?
  • Look at trends in vehicle registrations
  • Look at proportions of types of newly registered vehicles over time – why has the percentage of petrol cars being registered increased from 2015 to 2020?
  • Do some calculations to show that the number of petrol cars being registered has decreased from 2015 to 2020.
  • Reflect upon the implications for misleading representations of data
  • Consider the implications of the ban on new petrol and diesel cars by 2030 – what affect will this ban have on the proportions of car types being registered?
  •  What questions does the increase in electric vehicles raise?

Mathematical opportunities offered

  • Interpretation of data, statistics, graphs, infographics in context
  • Critiquing graphs
  • Reading scales
  • Calculating percentages
  • Exploring proportions of quantities over time
  • Making conjectures about future proportions given available data
  • Analysing and comparing data in order to develop and present a conclusion.
Download the resources
  1. Session plan
  2. Presentation
  3. Student sheet

Core Maths – Extreme Weather

Resource produced in collaboration with MEI

Brief overview of session ‘logic’

  • Do reports of extreme cold weather provide evidence that global warming is not happening?
  • Show the New York Times graphs of summer temperature distributions for the Northern Hemisphere for different periods.
  • Interrogate/critique these graphs
  • The distributions of temperatures are approximately Normal distributions and the mean and standard deviation both increase as the time period becomes more recent.
  • Use the dynamic bell curve to calculate probabilities of different temperatures in different time periods.
  • Despite the mean temperature increasing, the standard deviation also increasing means that the probability of extreme low temperatures increases.
  • Normal distributions and bell curves can explain a higher frequency of extreme cold weather despite global warming.

Mathematical opportunities offered

  • Interpretation of data, statistics, graphs, infographics in context
  • Critiquing graphs
  • Reading scales
  • Using standard form to write very large or very small numbers
  • Fitting a Normal distribution or bell curve to a graph
  • Exploring the effect of adjusting mean and standard deviation on a bell curve
  • Understanding that probabilities can be represented and calculated using areas
  • Analysing and comparing data in order to develop and present a conclusion.

Key Stage 3 – Extreme Weather

Resource produced in collaboration with MEI

Brief overview of session ‘logic’

  • Do reports of extreme cold weather provide evidence that global warming is not happening?
  • Show the New York Times graphs of summer temperature distributions for the Northern Hemisphere for different periods.
  • Interrogate/critique these graphs
  • The distributions of temperatures are approximately Normal distributions and the mean and standard deviation both increase as the time period becomes more recent.
  • Use the dynamic bell curve to calculate probabilities of different temperatures in different time periods.
  • Despite the mean temperature increasing, the standard deviation also increasing means that the probability of extreme low temperatures increases.
  • Normal distributions and bell curves can explain a higher frequency of extreme cold weather despite global warming.

Mathematical opportunities offered

  • Interpretation of data, statistics, graphs, infographics in context
  • Critiquing graphs
  • Reading scales
  • Using standard form to write very large or very small numbers
  • Fitting a Normal distribution or bell curve to a graph
  • Exploring the effect of adjusting mean and standard deviation on a bell curve
  • Understanding that probabilities can be represented and calculated using areas
  • Analysing and comparing data in order to develop and present a conclusion

Key Stage 3 – Trees and Carbon Capture

Resource produced in collaboration with MEI

Brief overview of session ‘logic’

  • Why trees are good
  • How much carbon do trees capture and store?
  • How does the amount of carbon captured and stored by a tree change during its lifecycle?

Mathematical opportunities offered

  • Interpretation of data, statistics, graphs, infographics in context
  • Critiquing graphs
  • Analysing and comparing data in order to develop and present a conclusion
  • Making assumptions
  • Making predictions
  • Reading scale
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