Speed: Multiplication and Division സമയവും വേഗവും
So far negative numbers told us where a point is and which way it moved. Now we let the point move with a speed for a certain time. To describe every case with the single equation s = vt, we must decide what (−10) × 3, 10 × (−2) and (−10) × (−2) mean – and then division follows.
- At a constant speed, distance = speed × time: 10 m/s for 4 s gives 40 m.
- Positions right of O are positive, left of O negative; x + (−y) = x − y; x − (−y) = x + y.
- For positive numbers, order of multiplying does not matter (5 × 3 = 3 × 5) and (x + y)z = xz + yz.
1. Speed and distance വേഗം
A point covers 50 metres in 5 seconds without speeding up or slowing down: its speed is 50 ÷ 5 = 10 metres per second (10 m/s). At the same speed it covers 100 m in 10 s. (The fastest 100 m ever run took 9.58 seconds – see the side box.)
Usain Bolt of Jamaica ran 100 m in 9.58 seconds in Berlin in 2009 – the fastest time ever recorded. But he did not run at one steady speed. The time for each 20 m was:
| Distance (m) | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 |
|---|---|---|---|---|---|
| Time (s) | 2.89 | 1.75 | 1.67 | 1.61 | 1.66 |
He was slowest at the start (getting going) and fastest between 60 and 80 m: 20 ÷ 1.61 ≈ 12.4 m/s. In this chapter our points move at a constant speed.
2. Left or right: s = vt and s = −(vt)
A point starts at O and moves at 10 m/s for 3 s. Moving right it reaches 10 × 3 = 30. Moving left it reaches −30 = −(10 × 3). With v for speed, t for time and s for the position reached:
moving right: s = vt moving left: s = −(vt)To merge these into one equation, write a leftward speed as a negative number: 10 m/s to the left is v = −10. Then we need a meaning for (−10) × 3.
For positive numbers, 10 × 3 means 3 lots of 10: 10 + 10 + 10 = 30. Use the same meaning: (−10) × 3 = (−10) + (−10) + (−10) = −30. Exactly the position reached! So s = vt works for both directions.
3. Two seconds before: negative time
Now we start our stopwatch at the moment the point passes O, still at 10 m/s. Two seconds after, a rightward point is at 20. Two seconds before, it had not yet reached O: it was 20 m to the left, at −20. The textbook table:
| Speed v | Time t | Position reached | s = vt ? |
|---|---|---|---|
| 10 (right) | 2 (after) | 20 | 20 = 10 × 2 |
| −10 (left) | 2 (after) | −20 | −20 = (−10) × 2 |
| 10 (right) | −2 (before) | −20 | −20 = 10 × (−2) |
| −10 (left) | −2 (before) | 20 | 20 = (−10) × (−2) |
Times after we start watching are positive, times before are negative. Rows 1 and 2 already work. Row 3 needs 10 × (−2), and row 4 needs (−10) × (−2).
4. Why 10 × (−2) = −20
For all positive numbers the order of multiplying does not matter: 5 × 3 = 3 × 5, ½ × ¼ = ¼ × ½, √2 × √3 = √3 × √2. Keep that rule: 10 × (−2) = (−2) × 10, and (−2) × 10 is ten lots of −2, which is −20. So 10 × (−2) = −20, and row 3 fits s = vt.
5. Why (−10) × (−2) = 20 – the distributive law
For the last row to fit, (−10) × (−2) must be 20. The textbook gives a reason based on arithmetic itself, not just on the motion.
x, y ഏത് അധിസംഖ്യകളായാലും – for any positive numbers x and y:
(−x)y = x(−y) = −(xy) (−x)(−y) = xyA negative times a positive is negative; a negative times a negative is positive. With these meanings, the position reached is s = vt for every speed and every time, whatever their signs.
Negative speed = moving left; negative time = before we started watching (the point was then on the other side of O from where it is heading).
Read the column from top to bottom: each step down subtracts (or adds) the same amount. The pattern only continues smoothly if a negative times a negative is positive.
| × | positive | negative |
|---|---|---|
| positive | positive | negative |
| negative | negative | positive |
(−1) × (−1) = 1, so (−1)² = 1. Multiply by −1 again: (−1)³ = −1. The powers of −1 swing between 1 and −1: for any natural number n, (−1)2n−1 = −1 (odd power) and (−1)2n = 1 (even power).
6. Division of negative numbers ന്യൂനഹരണം
For positive numbers, division is the reverse of multiplication: 6 ÷ 2 = 3 because 2 × 3 = 6 (and 6 ÷ 3 = 2 because 3 × 2 = 6). Use the same idea with negative numbers.
x, y ഏത് അധിസംഖ്യകളായാലും – for any positive numbers x and y (writing x ÷ y as x/y):
(−x)/y = x/(−y) = −(x/y) (−x)/(−y) = x/yThe signs work exactly as for multiplication: different signs give a negative answer, the same signs a positive one.
A ball thrown straight up at 49 m/s slows down by 9.8 m/s every second. After 5 s its speed is 49 − 5 × 9.8 = 0 and it starts to fall, speeding up by 9.8 m/s each second: at 7 s it falls at 2 × 9.8 = 19.6 m/s. Without negative numbers we need two equations: v = 49 − 9.8t for t < 5, and v = 9.8t − 49 for t > 5. With upward speed positive and downward speed negative, one equation does it all: v = 49 − 9.8t.
A tank losing 3 litres a minute has a "rate" of −3 L/min; 4 minutes ago it held 4 × 3 = 12 litres more, and (−3) × (−4) = 12 says exactly that. A shop losing ₹500 a day for 6 days has a total of (−500) × 6 = −₹3000. Sharing a debt of ₹600 among 3 people: (−600) ÷ 3 = −200 each.
7. Check yourself
- Leftward speeds and times before we start watching are negative. Then position reached = speed × time: s = vt always.
- (−x)y = x(−y) = −(xy): (−10) × 3 = −30 (repeated addition), 10 × (−2) = −20 (order does not matter).
- (−x)(−y) = xy: (−10) × (−2) = 20, chosen so that (x + y)z = xz + yz keeps working.
- (−1)ⁿ is −1 for odd n and 1 for even n.
- Division reverses multiplication: (−x)/y = x/(−y) = −(x/y), (−x)/(−y) = x/y.