Northern NSW da Vinci Decathlon SAMPLE An academic gala day for years 7 and 8

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beholder; and among the great features of Mathematics the certainty of its demonstrations is

1 Northern NSW da Vinci Decathlon SAMPLE An academic gala day for years 7 and 8 Mathematics Challenge Among all the studies of natural causes and reasons Light chiefly delights the beholder; and among the great features of Mathematics the certainty of its demonstrations is what preeminently (tends to) elevate the mind of the investigator Leonardo da Vinci. Session 1 Team Number

Mathematics of Light and Colour Mathematics of Light by Chin Chih Yang In this section we investigate some aspects of the mathematics of light and colour.

2 Mathematics of Light and Colour Mathematics of Light by Chin Chih Yang In this section we investigate some aspects of the mathematics of light and colour. A note on units 8 Speed of light = kilometres /second = 3 10 metres/second Use 1 year = 365 days. Please show all working as marks will be awarded for thinking! You can use a calculator where necessary. 2

3 Question 1 The Light Year (i) What is meant by the term light year? 1 mark (ii) How far does light travel in a year? Give your answer to 3 significant figures in scientific notation ( standard form). (iii) Suppose that a child is born on Earth in the year You are on an imaginary planet that is 94.6 trillion kilometres away from Earth and looking through a very high-powered telescope and you witness this child's third birthday party. How old, in years, is that child on Earth at the time you are watching the child's third birthday party? Explain your answer. 3

Question 2 Pentagonal Colours In how many different ways can I colour the five edges of a pentagon red, blue and green so that no two adjacent edges are the same colour?

4 Question 2 Pentagonal Colours In how many different ways can I colour the five edges of a pentagon red, blue and green so that no two adjacent edges are the same colour? 3 marks Reflections and rotations are to be accepted as different solutions. ( The same problem could be more practically understood by imagining painting a room with 5 walls so that no two adjacent walls will be of the same colour. (A tree diagram may be helpful.) 4

5 Question 3 Travelling at speeds approaching the speed of light Task developed by Dave Sedgman, Knox Grammar School, 2013 When objects start to travel at very large speed approaching that of the speed of light some very unusual things happen. The mass of the object increases according to the formula. M = M 0 v 1 c 2 2 Where M is the mass (kg) of the object travelling at speed v metres per second. M is the original mass (kg) of the object when it is stationary. 0 c is the speed of light in metres per second. (i) Show that when the object is travelling at three quarters of the speed of light that the mass M ( kg) of the object can be given as 4M M = 7 0 (ii) Show that this represents an increase in mass, compared to stationary, of approximately 51%. 5

Question 4 Lenses, Focal Length, Aperture and Camera f stops The focal length of a lens is the distance between the optical centre of the lens and the points where a clear image is formed.

6 Question 4 Lenses, Focal Length, Aperture and Camera f stops The focal length of a lens is the distance between the optical centre of the lens and the points where a clear image is formed. This is shown in figure 1 below. The focal length of modern camera lenses and telescopes is usually measured in millimetres. Focal length is very important in photography. Camera lenses each have a specific focal length. These range from 18 mm to over 500 mm Adjustable Camera Apertures - f-stops Unlike telescopes, you can adjust the aperture (or diameter) of many camera lenses as shown in Figure 2 using a ring on the lens or an electronic button. This is one of the ways to get the proper amount of light for the photograph. Figure 2: Adjustable Aperture Lens. (Changing the diameter of the opening of the lens to vary the amount of light that can pass through.) Large diameter (aperture) means a small f-stop ( f1.4) that allows maximum amount of light to pass through the lens into the camera. Small diameter (aperture) means a large f-stop( f16) that allow very little light into the camera. 6

7 Question 4 continued Task developed by Dave Sedgman, Knox Grammar School, 2013 (i) Photographers use a formula to find the f-stop. f stop = focal length of lens diameter of lens Find the diameter of the aperture of a lens which has a 28 mm focal length which has an f-stop = 2. (ii) Find the area of light that will pass through this aperture as described in part (i). Give your answer to the nearest mm 2. (iii) f stop Area of aperture (mm 2 ) 1 mark The f stop column is an example of a geometric series, where each successive number is a fixed multiple of the previous number. This is called the common ratio. Which irrational number do you think this common ratio is for the f stop column? 7

Question 5 Image formation by Lenses Task developed by Dave Sedgman, Knox Grammar School, 2013 The thin lens equation relates the object distance and the image distance to the focal length of a lens

8 Question 5 Image formation by Lenses Task developed by Dave Sedgman, Knox Grammar School, 2013 The thin lens equation relates the object distance and the image distance to the focal length of a lens as follows: = + or focal length object distance image distance = + f d0 d i (i) What focal length must a convex lens have to produce an image distance of 6cm when an object is placed 6cm from the lens? (ii) Use the lens formula to calculate the image distance when an object is placed 10 cm from a convex lens with a focal length of 8cm. 8

Question 6 Computer generated colours Task developed by Dave Sedgman, Knox Grammar School, 2013 Computers use 256 different values for each of the three primary colours.

9 Question 6 Computer generated colours Task developed by Dave Sedgman, Knox Grammar School, 2013 Computers use 256 different values for each of the three primary colours. This includes just about every possible shade of every colour imaginable, more than you could possibly name. Why do computers make so many colours? The answer is that, as far as the computer is concerned, every different combination of Red, Green, and Blue counts as a different colour, even if you couldn't tell the difference just by looking. Calculate the total number of colour combinations possible on a computer screen. 9

10 Question 7 Colour Blindness Task developed by Dave Sedgman, Knox Grammar School, 2013 "Colour blindness" is an inability to see certain colours. Most colour blindness is a result of a lack of one or more of the types of colour cones in the retina. About 10% of males have some colour perception defect, but it is practically non-existent in females. The most common colour perception defects are for red or green or both. A group consists of 200 people with equal number of males and females and one person is selected at random by a blindfolded person. Write down the probability P(CBM) that the person selected at random will be a blind male. colour 10

give magenta: (See diagrams above) white - green = blue + red = magenta white - red = blue +

11 Question 8 Colour Arithmetic We can represent what happens by means of "colour arithmetic". For example, if you take away green from white, you are left with blue and red, which combine to give magenta: (See diagrams above) white - green = blue + red = magenta white - red = blue + green = cyan white - blue = green + red =??? What colour is represented in the last line above? 1 mark 11

12 Question 9 Intensity of Light at a Depth The percent I (d) remaining of the original intensity of light that is available at a depth d metres below the surface of the water in a lake is given by the equation, Id ( ) = 100 (2.72) 0.95 d What percentage of the light, to the nearest tenth of a percent, is remaining 2 metres below the surface of the lake? 12

13 End of Challenge Task developed by Dave Sedgman, Knox Grammar School, 2013 Chess Challenge White to play and checkmate black in 5 moves White to play and checkmate black in 1 move 13

14 White to play and checkmate black in 2 moves White to play and checkmate in 2 moves 14

CHAPTER 18 REFRACTION & LENSES

Physics Approximate Timeline Students are expected to keep up with class work when absent. CHAPTER 18 REFRACTION & LENSES Day Plans for the day Assignments for the day 1 18.1 Refraction of Light o Snell