STEM B-29 LEARNING RESOURCESUPPORTING MATHS TEACHER NOTES

Syllabus Links

Number Page 2

Algebra Page 3

Ratio, proportion and rate of change Page 4

Geometry and measure Page 4

Sample questions Page 5

These notes are designed to support teachers of KS3 & GCSE Mathematics linking to the STEM B-29 learning resource: How did the B-29 Superfortress become the most advanced bomber of the Second World War?

These notes indicate relevant topic areas and give some exemplar tasks that reflect examination expectations, in line with schemes from the major examination boards. Topic references are taken from the National Curriculum GCSE mathematics. Questions have been written that use the data relating to the B-29 and B-17. They can be used as is or modified to meet particular needs of a class or lesson.

Contents

About this resource

The standard data for these two bombers allow for a variety of good arithmetic.

N2 Four rules with large numbers

N2 Place value

N7 Calculate with roots, and with integer indices

N9 Calculate with and interpret standard form

N13 Manipulation of values expressed in units such as mass, length and time

N14 Estimation of converted figures and validity of estimation used

N15 Rounding of original and/or converted values to an appropriate degree of accuracy

N.B. Imperial units have been used in these notes as they were the units in use at the time of the aircraft’s construction.

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Number

One of the key features of the B-29 was the significant increase in wing area to allow it to carry much greater payloads. The Lift Equation permits a number of algebraic tasks.

N1 Use and interpret algebraic notation

N2 Substitute numerical values into formulae

N5 Rearrange formulae to change the subject, including use of formulae from other subjects

N6 Use algebra to support and construct arguments

N14 Simple kinematic problems involving distance, speed and acceleration

N15 Calculate gradients of distance-time graphs

Suggestion

Questions could also be arranged so that conversion between units has to take place before calculating values. However, metric units have been used for the sample tasks so that the gravitational constant is roughly 10 ms-2, making for clear questions that can be modified.The large values lend themselves to change into standard form.

Lift Equation L = 12 ρ v2 s CL

[The letters used in the question have been altered to match the simplified form]

Simplified formlift = constant x wing area x speed2

For the take-off point of a fully laden B-29 this constant handily approximates to 1. [at sea level and 15°C]

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Algebra

Units:Lift LAir density pVelocity of aircraft vthe wing area of the aircraft scoefficient of lift, relates to the type of wing and angle of attack

CL

R1 Change freely between related standard units (e.g. time, length, area, volume/capacity, mass)

R2 Scale factors, scale diagrams, maps

R9 Variety of tasks with percentages

R12 Compare lengths and areas and volumes using scale factors

This topic is tested throughout the questions below.

G14 Use standard units of measure and related concepts

The wings of WWII aircraft are relatively straightforward shapes and can be approximated to trapezia allowing calculation and comparison of areas, and consideration of scale factors.

G16 Know and apply formulae to calculate area of trapezia

G19 Apply the concepts of congruence and similarity, including the relationships between lengths and areas in similar figures

Route maps for B-29 missions lend themselves well to scale drawing/bearings questions which can be extended to cover angle properties, trig and Pythagoras.

G1 Use the standard conventions for labelling; draw diagrams from written description

G2 Use ruler and compass to construct given figures

G15 Measure line segments and angles in geometric figures, including interpreting maps and scale drawings and use of bearings

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Ratio/proportion/rate of change

Geometry and measure

B-29 Specification B-17 B-29Crew 10 11

Engines 4 x 1200 hp 4 x 2200 hp

Wingspan 103 ft. 9 in 142 ft. 3 in

Length 74 ft. 4 in 99 ft.

Mass (empty) 36135 lb 71500 lb

Mass (fully laden) 72000 lb 140000 lb

Bomb load (typical) 4-5000 lb 5-12000 lb

Bomb load 17600 lb(rare overload max.)

20000 lb(maximum short range)

Speed (cruising) 180 mph 230 mph

Speed (maximum) 300 mph 399 mph

Service ceiling 35000 ft. 39650 ft.

Range 2000 miles(with 4000 lb load)

4000 miles(with 5000 lb load)

No. built (by end of WWII) 13000 3900

1. For this question, use the values in the data boxes for the B-17 and B-29 bombers.

Remembering that there are 12 inches (in) in 1 foot (ft.),

1.1. Calculate the wingspan of a B-17 in metres.1.2. Calculate the wingspan of a B-29 in metres.1.3. Work out the difference between the wingspans.1.4. Express the difference as a percentage of the wingspan of a B-17.1.5. Convert the lengths to metres then express the difference in length between the two

bombers as a percentage of the length of the shorter aircraft.1.6. Comment on the two values obtained in parts 1.1 and 1.5.

2. Using the values in the data boxes for the B-17 and B-29 bombers:

2.1. Calculate the combined mass of one of each bomber in pounds (lb).2.2. Give the value of the left hand digit of your answer to part 2.1.2.3. Given that 1 kg = 2.2 lb convert your answer to part 2.1 to kilograms correct to 3

significant figures.2.4. Change your answer to part 2.3 into tonnes.2.5. Stating approximations made, given estimate of the mass of all B-29s ever built.

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Sample questions

3. B-29 bombers have a maximum fuel capacity of 9400 US gallons.

3.1 Given that 1 US gallon = 3.8 litres, find the fuel capacity of the bomber in litres.

A ‘Wing’ of 20 B-29 bombers are being prepared for a mission requiring maximum fuel. The WWII Chevrolet fuel service truck, called a bowser, carries 750 US gallons.

3.2 Work out the total fuel needed in litres.3.3 Convert this answer to cm³; give your answer in standard form.3.4 How many bowsers are needed altogether to refuel the wing?

The average fuel consumption of a B-29 is 475 US gallons per hour.The pilot’s handbook states that fuel required for a given flight has a 10% safety margin.

3.5 Work out the longest time a B-29 can fly.3.6 Give TWO reasons why the handbook gives the figure of 10%.

Forces acting on a B-29, © IWM (FRE 11935)

lift = wing area x speed2 [Newtons]weight = mass x 10 [Newtons]

4. A fully laden B-29 needs to reach 60 m/s to take off.The wing area of a B-29 is 160 m2.

4.1. Calculate the lift (in Newtons) at take-off.

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4.2. Rewrite the lift in standard form.

This B-29 has a mass of 56 000 kg.

4.3. Work out the weight (in Newtons) of the fully laden B-29.4.4. Rewrite the weight in standard form.4.5. What do you notice about the values for weight and lift at the point of take-off?4.6. What will happen if the aircraft can only reach 59 m/s?

We can re-write the lift equation using symbols so that:

l = A x s24.7. Re-arrange this equation so that s is the subject.

The lift equation above is an approximation. The real equation is:

L = dAs 2 C L 2 2where d is air density and CL is the coefficient of lift; air density is 1.225 kg/m3

4.8. Re-arrange the equation to make CL the subject.4.9. Work out the coefficient of lift for the B-29 at the point of take-off; lift and weight can be

assumed to be equal at this point.

At higher altitudes the air density will be lower.

4.10. What will happen to the lift? 4.11. How will that affect the point of take off?4.12. How can the pilot compensate for this?

5. Here is the speed-time graph for the first five minutes of a B-29 sortie from brakes off.

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Spee

d in

met

ers

per s

econ

d

Flight time in seconds

Speed-time graph depicting the first five minutes of a B-29 sortie from brakes off

5.1. The aircraft takes off when it reaches at 60 m/s. How long does this take?5.2. Calculate the acceleration while on the ground.5.3. Calculate the acceleration while on the ground.

The B-29 was found to have overheating problems on take-off so pilots were instructed:A) climb steadily while the undercarriage comes up and the flaps move back into placeB) continue climbing to 500 feetC) level out and accelerate to the required speed before climbing to altitude.

5.4. Label the sections of the graph where A, B and C occur.5.5. What happens to the gradient of the graph as it moves between sections A, B and C?5.6. Explain what this means in terms of acceleration and suggest why this happens.5.7. What climbing speed does the pilot reach?5.8. How long does it take from the point of take-off to starting the climb to altitude. Give your

answer in minutes and seconds.5.9. How far does the aircraft travel when climbing to altitude?

6. Here are the wing sections from both the B-17 and the B-29. The trapezia are useful approximations to the wings of each plane.

B17 and B29 wing section diagrams © IWM

6.1. Calculate the area of each trapezium and hence the wing area of each aircraft.6.2. How many times larger is the B-29 wing area than the B-17 wing area?6.3. How many times longer is the B-29 wing than the B-17 area?

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6.4. If the B-17 wing had the same wing area as the B-29 but was not any longer, how wide would the wing be at its root, assuming both ends increased in proportion?

6.5. What does this tell you about the different designs of each wing?

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7. Here is a map of industrial targets in Japan that were attacked from the Mariana Islands.The B-29s would fly to Iwo Jima to collect fighter escorts then travel to their target.

Industrial targets in Japan © IWM

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7.1. Plot a course from the northern Mariana Island, Saipan, via Iwo Jima to attack Tokyo, then Nagasaki, returning directly to Saipan.

7.2. Measure and label all distances and bearings, remembering to construct a North line at each location.

Before flying such a mission, planners would work out a detailed breakdown of each stage so that fighter escorts are ready at the correct time and fuel/bomb loads can be worked out beforehand. Write a stage by stage plan for the raid, with distances, bearings and timings for each section, allowing 5 minutes extra over each target, leaving Saipan at 1000 hrs.

7.3. Calculate the total distance covered.7.4. Using an average speed of 350 kph, how long will the sortie take?7.5. With a fuel burn rate of 1800 litres an hour, work out how much fuel will be needed for

the sortie; remember to add on a 10% margin of error.7.6. Express this value as a percentage of the maximum fuel capacity of the aircraft.7.7. 100 litres of aviation fuel weighs 72 kg. Work out the mass of fuel needed for the flight.

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