Directions, Number Line and Signs പോസിറ്റീവ് / നെഗറ്റീവ്
The chapter opens with a child walking between electric posts on a village road. It asks: how far did the child walk, where is the child now, and in which direction? To answer such questions you need four simple tools: the directions, a number line to mark positions, + and − signs for the two directions along a line, and the difference between a straight path and a curved path. This page builds all four from scratch.
- Counting, adding and subtracting whole numbers.
- Nothing else: negative numbers are explained below.
1. The four directions
There are four main directions: north (വടക്ക്), east (കിഴക്ക്), south (തെക്ക്) and west (പടിഞ്ഞാറ്). The sun rises in the east and sets in the west. If you stand facing the sunrise, north is on your left and south is on your right. A magnetic compass needle always points north–south.
On every map (and in your textbook's figures) north is drawn at the top. So east is to the right, west to the left and south at the bottom. Halfway between two directions we say north-east, south-east, south-west or north-west.
Turning right moves you clockwise round the compass: north → east → south → west → north. An “about turn” makes you face the opposite direction.
Kerala is a long, narrow state. The Arabian Sea is along its west, and the Western Ghats run along its east. That is why the sun sets over the sea at Kozhikode or Kovalam beach. Most rivers, such as the Bharathapuzha and the Periyar, flow roughly from east to west, from the hills to the sea. If you travel from Thiruvananthapuram to Kasaragod, you are going roughly north.
2. Positions on a number line
A road, a railway track or a straight corridor can be thought of as a number line. First choose a starting mark and call it 0. This is the origin (the reference point). Then every place on the line gets a number: how many metres it is from the origin.
- Places to the right (east) of the origin get positive (പോസിറ്റീവ്) numbers: +10 m, +40 m …
- Places to the left (west) of the origin get negative (നെഗറ്റീവ്) numbers: −10 m, −40 m …
The number of a place is called its position. The place where a motion starts is the initial position (ആദ്യസ്ഥാനം) and the place where it ends is the final position (അന്ത്യസ്ഥാനം).
The posts are 40 m apart, as in the textbook's Figure 2.1. Change the origin: every position number changes, but the gap between two posts stays the same. Gap = (larger position) − (smaller position).
The kilometre stones along the NH 66 work like a number line. A stone that reads “Kochi 25” tells you that you are 25 km from Kochi, which is the origin for that stone. A stone a little further on reads 26, then 27, as you move away.
3. The + and − directions
Along a straight line there are only two directions. Physics uses a sign to say which one:
Choose one direction as positive (+), usually east, right, forward or up. The opposite direction, west, left, backward or down, is then negative (−). Once you have chosen, do not change it in the middle of a question. In this guide, east (→) is + unless a question says otherwise.
| In words | With a sign (east / up = +) |
|---|---|
| 40 m towards the east | +40 m |
| 40 m towards the west | −40 m |
| a ball going up at 5 m/s | +5 m/s |
| a coconut falling down at 10 m/s | −10 m/s |
| back where you started | 0 m |
The number tells the size (how much) and the sign tells the direction. The size without its sign is called the magnitude: the magnitude of −40 m is 40 m. So “−40 m” is not “less than nothing”; it simply means “40 m the other way”.
4. Adding and subtracting signed numbers
Think of each signed number as a move along the line: + moves you right (east), − moves you left (west). To find where you end up, start at your initial position and do the moves one after another. That is just adding signed numbers:
Green hops go east (+), red hops go west (−). The total length walked adds up all the hops without signs. The final position adds the signed moves. They are different as soon as you turn back. You will meet these two ideas again as distance (ദൂരം) and displacement (സ്ഥാനാന്തരം) on page 5.
Rules for signs
| Rule | Example | On the number line |
|---|---|---|
| Adding a negative = subtracting | 30 + (−50) = 30 − 50 = −20 | from 30, go 50 left |
| Subtracting a negative = adding | 30 − (−20) = 30 + 20 = 50 | the gap from −20 up to 30 is 50 |
| Change = final − initial | from +150 to −50: −50 − 150 = −200 | 200 to the left |
| (+) × (+) = + and (−) × (−) = + | (−5) × (−4) = 20 | same signs give + |
| (+) × (−) = − and (−) × (+) = − | (−5) × 4 = −20 | different signs give − |
| Division follows the same sign rules | (15 − 20) ÷ 10 = −5 ÷ 10 = −0.5 | different signs give − |
In this chapter you will multiply velocities and accelerations by time. A motorbike slowing down from 20 m/s to 15 m/s in 10 s has acceleration (15 − 20) ÷ 10 = −0.5 m/s². The minus sign tells you it is slowing down (page 8). Get the sign wrong and the whole answer is wrong.
Worked example 1 · Change in position
Q. Taking east as positive, an autorickshaw starts at the position +150 m and stops at the position −50 m on a straight road. Find the change in its position and say in which direction it moved.
Change = final position − initial position. Watch the signs: subtracting +150 is the same as taking away 150.
Worked example 2 · Forward and back
Q. Starting at post B, a child walks 40 m east to post C, another 40 m east to post D, then turns back and walks 40 m west to post C. Taking B as the origin and east as positive, find (a) the total length walked and (b) the child's final position.
Write the moves with signs: +40, +40, −40. For the total length, ignore the signs. For the final position, add the signed moves to the start (0).
5. Straight paths and curved paths
The route an object actually follows is its path (സഞ്ചാരപാത). A path can be a straight line, a line with turns, or a curve. A path along a circle is a circular path (വർത്തുളപാത). Two things to notice:
- Between two places, the straight line is always the shortest path. Any path with turns or curves is longer.
- On a straight path you keep the same direction of motion (ചലനദിശ). On a curved path your direction of motion changes at every point. At each instant you are heading along the curve, in the direction a ball would fly off if you let it go.
North is up. School is 240 m east and 180 m north of home, so the straight line between them is 300 m long (a 3 : 4 : 5 right triangle). The orange arrow shows the direction of motion. It never turns on the straight path, turns once at the corner, and turns all the time round the pond.
Length of the path you walk ≥ straight-line gap from start to end. They are equal only for a straight path in one direction. You will meet this again as a key idea on page 5.