Rearranging Formulas, Squares and Areas പരപ്പളവ്
The chapter's numericals need a handful of maths skills. You must turn a formula round to find the quantity you want, put numbers in carefully, square a number and find a square root (√1600 = 40), find the area of a rectangle, a triangle and a trapezium, and handle fractions like 1/24. None of it is hard once you have seen it done slowly. This page does exactly that.
- The units m, s, m/s and m/s² (page 2) and the sign rules (page 1).
- Multiplication tables; a calculator is not needed for the textbook numbers.
1. Turning v = s/t round
Velocity is displacement divided by time: v = s/t. Sometimes the question gives v and t and asks for s, or gives s and v and asks for t. You do not need three formulas to remember, just one and the formula triangle:
Cover the quantity you want. What is left shows the formula: s over v means s ÷ v; v beside t means v × t. (The slider of the unknown quantity is ignored: the answer is calculated from the other two.)
Worked example 1 · How long?
Q. A cyclist rides along a straight road at a uniform velocity of 5 m/s. How long does she take to cover 300 m?
Time is wanted. Cover t in the triangle: s over v, so t = s ÷ v.
2. Turning a = (v − u)/t round
Acceleration is the change in velocity divided by the time: a = (v − u)/t, where u is the initial velocity (ആദ്യപ്രവേഗം) and v the final velocity (അന്ത്യപ്രവേഗം). An equation is like a balance. Whatever you do to one side, do the same to the other side, and it stays true:
| Step | What we do to both sides | Result |
|---|---|---|
| start | a = (v − u) ÷ t | |
| 1 | multiply both sides by t | at = v − u |
| 2 | add u to both sides | v = u + at (the first equation of motion!) |
| 3 | from at = v − u, subtract at from both sides and add u to both sides | u = v − at |
| 4 | from at = v − u, divide both sides by a | t = (v − u) ÷ a |
The graph shows the velocity changing from u to v in t seconds. The slope of the line is the acceleration. A negative a means the line goes down: the object is slowing down.
Worked example 2 · How long to speed up?
Q. A car speeds up from 10 m/s to 30 m/s with a uniform acceleration of 4 m/s². How long does it take?
u = 10 m/s, v = 30 m/s, a = 4 m/s². Time is wanted: t = (v − u) ÷ a.
3. Substituting values in the right order
When you put numbers into a formula, work in this order (BODMAS): Brackets first, then Orders (squares), then Division and Multiplication, and last Addition and Subtraction. In s = ut + ½at², the little ² belongs to t only: square t first, then multiply by a and by ½.
Start: u = 3, a = 2, t = 4 gives 3 × 4 + ½ × 2 × 4² = 12 + ½ × 2 × 16 = 12 + 16 = 28 m. The red line shows the common mistake of squaring a × t together: ½ × (2 × 4)² = 32, which is wrong.
4. Squares and square roots
The square of a number is the number multiplied by itself: 4² = 4 × 4 = 16, 20² = 400. It is the area of a square with that side. The square root √ goes backwards: √400 asks “which number times itself makes 400?”. The answer is 20. In the equation v² = u² + 2as you first find v², then take the square root to get v.
| n | 2 | 3 | 4 | 5 | 6 | 8 | 10 | 15 | 20 | 25 | 30 | 40 | 60 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| n² | 4 | 9 | 16 | 25 | 36 | 64 | 100 | 225 | 400 | 625 | 900 | 1600 | 3600 |
√1600 = √(16 × 100) = √16 × √100 = 4 × 10 = 40. In the same way √3600 = 6 × 10 = 60, and 300² = 3² × 100² = 9 × 10,000 = 90,000.
Area of the square = n × n = n². Reading it backwards, the side = √(area).
Worked example 3 · Square, then square root
Q. Work out v from v² = u² + 2as when u = 10 m/s, a = 2 m/s² and s = 75 m.
Square u first: 10² = 100. Then 2 × 2 × 75. Add, and finally take the square root.
5. Areas: rectangle, triangle and trapezium
On page 10 you will find the displacement from the area (പരപ്പളവ്) under a velocity–time graph. Only three shapes ever appear:
The rectangle uses the left height for its height. The triangle starts from a point at the left and rises to the right height, just like a v–t graph that starts from rest. The trapezium is a rectangle (blue) plus a triangle (purple) on top, which is why its formula works.
Worked example 4 · Area of a trapezium
Q. A trapezium has parallel sides of 10 and 20 units, and they are 6 units apart. Find its area. (On a v–t graph this is a velocity rising from 10 m/s to 20 m/s in 6 s.)
Area = ½ × (a + b) × h with a = 10, b = 20 and h = 6. Or split it: rectangle 10 × 6 plus triangle ½ × 6 × 10.
6. Fractions and decimals
Some answers come out as fractions. To simplify a fraction, divide the top and the bottom by the same number. To change it into a decimal, divide the top by the bottom. Useful ones: ½ = 0.5, ¼ = 0.25, 5/18 ≈ 0.278.
Try 20 over 300 (you get 1/15, from “Let us assess” question 6) and 5 over 18 (already in its simplest form).
Worked example 5 · A small acceleration
Q. A calculation gives a = 25/600 m/s². Write it as a fraction in its simplest form and as a decimal to three places.
Both 25 and 600 can be divided by 25. Then divide 1 by the bottom number.
A long passenger train leaving Shoranur Junction speeds up very gently. Its acceleration is often less than 0.1 m/s², a small fraction such as 1/24 m/s². That is why it takes several minutes to reach full speed. An autorickshaw pulling away from a signal accelerates many times faster, but it has a much lower top speed.
7. Check yourself
You are ready for the chapter! Next: Distance and displacement ▶