Speed, Ratios and Powers of Ten പ്രകാശവേഗം
The textbook tells you that light travels at 3 × 10⁸ m/s in air, slower in water and glass, and that the refractive index is a ratio of two speeds. Then it asks you to work out things like 3 × 10⁸ ÷ 2.4. This page gives you every bit of arithmetic you need: what speed means, how to read and write huge numbers with powers of ten, how to divide them, why a ratio has no unit, how to put decimals like 1.33 and 1.52 in order, and what it means when “one goes up and the other goes down”.
- Multiplying and dividing simple numbers, and decimals like 2.5 or 1.33.
- Nothing else – everything else is built here.
1. Speed = distance ÷ time
Speed tells you how much distance something covers in one unit of time. If you walk 100 m in 50 s, you cover 2 m every second: your speed is 2 metres per second, written 2 m/s.
The unit of speed is a distance unit “per” a time unit: m/s (metres per second) in science, km/h (kilometres per hour) on a vehicle's speedometer. The symbol for speed is usually v.
More distance in the same time → more speed. The same distance in less time → more speed too. Try 100 m in 12.5 s: that is 8 m/s, a good school sprinter.
Worked example 1 · Speed of a sprinter
Q. At the sub-district sports meet, Arjun runs the 100 m race in 12.5 s. Find his average speed.
Speed = distance ÷ time. To divide by 12.5, multiply both numbers by 2: 100 ÷ 12.5 = 200 ÷ 25.
2. Powers of ten
Light is so fast that its speed is a very long number. To avoid writing lots of zeros, we use powers of ten. The small raised number (the power) tells how many 10s are multiplied together – which is the same as how many zeros follow the 1:
| Power | Means | Written out | In Indian words |
|---|---|---|---|
| 10¹ | 10 | 10 | ten |
| 10² | 10 × 10 | 100 | hundred |
| 10³ | 10 × 10 × 10 | 1 000 | thousand |
| 10⁵ | five 10s | 1 00 000 | one lakh |
| 10⁷ | seven 10s | 1 00 00 000 | one crore |
| 10⁸ | eight 10s | 10 00 00 000 | ten crore |
Each step up in the power makes the number 10 times bigger: the digits shift one place to the left and one more zero appears. Multiplying by 10⁸ moves the decimal point 8 places to the right.
In a vacuum (ശൂന്യത) – empty space – light travels at c = 3 × 10⁸ m/s = 300 000 000 m/s = 30 crore metres every second = 3 lakh kilometres per second. In air it is almost exactly the same. The letter c is always used for this speed.
The distance round the Earth is about 40 000 km. In one second, light could go round it about 300 000 ÷ 40 000 = 7.5 times. Sunlight takes 1.5 × 10¹¹ m ÷ 3 × 10⁸ m/s = 500 s (8 minutes 20 seconds) to reach Kerala, and moonlight only about 1.3 s. A KSRTC bus at 60 km/h would need almost 285 years to cover the distance to the Sun!
3. Writing numbers in scientific notation
Scientists write big numbers as a number between 1 and 10, multiplied by a power of ten. This is called scientific notation (or standard form). To change 225 000 000 into this form:
- Put a decimal point after the first digit: 2.25000000.
- Count how many places the point moved to the left: 8.
- Write the number with that power of ten: 2.25 × 10⁸.
Every place the point moves left divides the front number by 10, so the power of ten must go up by 1 to keep the value the same. Stop when exactly one digit (not 0) is in front of the point.
4. Dividing numbers with powers of ten
In this chapter you will divide 3 × 10⁸ by other numbers. There are only two cases:
Case 1 · divide by a plain number
Divide the front numbers; the power of ten stays.
Case 2 · both have the same power
The powers of ten cancel; divide the front numbers.
Dividing by a decimal: multiply both numbers by 10 (or 100) until the divisor is a whole number. 3 ÷ 2.4 = 30 ÷ 24 = 1.25. 3 ÷ 2.25 = 300 ÷ 225 = 4 ÷ 3 = 1.333… ≈ 1.33.
Case 1 gives a speed (it still has × 10⁸ and the unit m/s). Case 2 gives a plain number – a ratio – with no power of ten and no unit. Try b = 2.4, 1.44, 2 and 2.25.
Worked example 2 · Case 1
Q. Work out 3 × 10⁸ ÷ 2.4. (You will meet this sum on page 6, for the speed of light in diamond.)
Keep the 10⁸. Work out 3 ÷ 2.4 by multiplying both by 10: 30 ÷ 24.
Worked example 3 · Case 2
Q. Light travels at 3 × 10⁸ m/s in air and 2.25 × 10⁸ m/s in water. How many times faster is it in air than in water?
“How many times” means divide. Both speeds have × 10⁸ and m/s, so both cancel. 3 ÷ 2.25 = 300 ÷ 225.
5. A ratio has no unit
A ratio compares two quantities of the same kind by dividing one by the other. Because the units are the same on the top and the bottom, they cancel:
This is exactly why the refractive index (അപവർത്തനാങ്കം) you will meet on page 6 has no unit: it is the speed of light in vacuum divided by the speed of light in the material.
Writing a unit after a ratio, e.g. “refractive index = 1.5 m/s”. A ratio of two speeds is just a number: 1.5.
6. Putting decimals in order
Later you will compare refractive indices such as 1.33, 1.44, 1.47, 1.52 and 2.42. To compare two decimals, line up their decimal points and compare place by place from the left. Put in zeros at the end if one is shorter: 1.5 = 1.50, so 1.5 is bigger than 1.47 (because 5 tenths beats 4 tenths).
The top line shows everything; the bottom line zooms in between 1.30 and 1.65, where the liquids and glasses crowd together. Glycerine and sunflower oil both have 1.47 – they are equal.
7. “One goes up, the other goes down” – inverse relation
If you travel 80 km from Kochi to Thrissur at 40 km/h you need 2 hours; at 80 km/h only 1 hour. Doubling the speed halves the time. When one quantity goes up exactly as the other goes down, so that their product stays the same (40 × 2 = 80 × 1 = 80), we say they are inversely proportional.
The same happens with light. In the next pages you will learn that speed in a material = 3 × 10⁸ ÷ (refractive index). A bigger refractive index means a smaller speed.
The curve always falls: as n goes up, 3 × 10⁸ ÷ n comes down. At n = 2 the answer is exactly half of 3 × 10⁸. The product n × speed is always 3 × 10⁸.