Hello, and welcome to today's mathematics lesson. Today, we are diving into Speed, Distance and Time. By the end of this lesson, you will understand how to calculate speed, distance, and time, how to convert between different units, and how to solve problems involving average speed, relative speed, and trains. Let us begin.
We start with the most fundamental concept: speed.
Speed tells us how fast something is moving. More precisely, the distance covered by a moving object in unit time is called its speed.
This gives us our first formula.
Speed equals distance divided by time. In symbols: Speed = Distance/Time.
Now, units matter enormously here. If distance is measured in metres and time in seconds, then speed is in metres per second, written as m s⁻¹. If distance is in kilometres and time in hours, speed is in kilometres per hour, written as km h⁻¹.
Other combinations work too. Centimetres per second, metres per minute, kilometres per minute — the pattern is always distance unit per time unit.
Let us work through an example together. A boy walks 300 metres in 2 minutes. We want his speed in three different units.
First, in metres per minute: speed equals 300 metres divided by 2 minutes, which gives 150 metres per minute.
Second, in metres per second: we need consistent units, so convert 2 minutes to 120 seconds. Speed equals 300 metres divided by 120 seconds, which simplifies to 2.5 metres per second.
Third, in kilometres per hour: convert 300 metres to 0.3 kilometres, and 2 minutes to one-thirtieth of an hour. Speed equals 0.3 divided by one-thirtieth, which equals 0.3 times 30, giving 9 kilometres per hour.
From our basic formula, we can derive two more essential relationships.
First, distance. Since speed equals distance over time, we can rearrange to get: distance equals speed multiplied by time.
For instance, if a car travels at 10 kilometres per hour for 2 hours, the distance covered is 10 times 2, which equals 20 kilometres.
Be careful with units. If speed is 4 kilometres per minute and time is 2 hours, first convert 2 hours to 120 minutes. Then distance equals 4 times 120, giving 480 kilometres.
Second, time. Rearranging again: time equals distance divided by speed.
If an object covers 800 metres at 40 metres per second, the time taken is 800 divided by 40, which equals 20 seconds.
Or, if a man covers 4 kilometres at 12 kilometres per hour, time equals 4 over 12 hours, which is one-third of an hour, or 20 minutes.
Now, let us tackle unit conversions — a skill you will use constantly.
To convert from kilometres per hour to metres per second, remember this: 1 kilometre per hour equals 1000 metres divided by 3600 seconds, which simplifies to 5/18 metres per second.
So, multiply by 5/18 to convert kilometres per hour to metres per second. For example, 72 kilometres per hour becomes 72 times 5/18, which equals 20 metres per second.
To convert the other way, from metres per second to kilometres per hour, multiply by 18/5, which is 3.6. So 15 metres per second becomes 15 times 18/5, which equals 54 kilometres per hour.
Let us see this in action. A car covers 375 metres in 25 seconds. First, find its speed in metres per second: 375 divided by 25 equals 15 metres per second. Convert to kilometres per hour: 15 times 18/5 equals 54 kilometres per hour.
Now, if this same car covers 525 metres at the same speed, how long does it take? Time equals 525 divided by 15, which equals 35 seconds.
Next, we turn to average speed.
When a journey has different parts with different speeds, we need the overall average speed.
Average speed equals total distance covered divided by total time taken.
Imagine a car travels from A to P, 50 kilometres in 2 hours, then P to Q, 170 kilometres in 3 hours, then Q to B, 20 kilometres in 1 hour. Total distance is 50 plus 170 plus 20, which equals 240 kilometres. Total time is 2 plus 3 plus 1, which equals 6 hours. Average speed is 240 divided by 6, giving 40 kilometres per hour.
Here is a crucial warning: average speed is NOT the average of the speeds. If a car covers 250 kilometres at 50 kilometres per hour, that takes 5 hours. Then 120 kilometres at 40 kilometres per hour takes 3 hours. Then 50 kilometres at 25 kilometres per hour takes 2 hours. Total distance is 420 kilometres, total time is 10 hours, so average speed is 42 kilometres per hour — not the simple average of 50, 40, and 25.
Now, relative speed — this describes how fast the gap between two moving objects changes.
When two objects move in opposite directions, their relative speed is the sum of their individual speeds. The distance between them either shrinks or grows by this combined amount each hour.
When two objects move in the same direction, their relative speed is the difference of their speeds. The faster one either catches up to or pulls away from the slower one at this rate.
Consider two boys starting together: one walks at 8 kilometres per hour, the other at 9 kilometres per hour. After 5 hours walking in the same direction, how far apart are they? Relative speed is 9 minus 8, which equals 1 kilometre per hour. Over 5 hours, the gap becomes 5 kilometres.
If they walk in opposite directions, relative speed is 9 plus 8, which equals 17 kilometres per hour. After 5 hours, they are 85 kilometres apart.
Finally, we apply these ideas to trains — a classic application.
When a train crosses a pole or a stationary person, the distance covered equals just the length of the train.
When a train crosses a platform, bridge, or tunnel, the distance covered equals the length of the train plus the length of the platform.
Let us work through this carefully. A 540 metre train runs at 81 kilometres per hour. First, convert the speed: 81 times 5/18 equals 45/2 metres per second, which is 22.5 metres per second. Time to cross a pole equals 540 divided by 22.5, which equals 24 seconds.
Here is a more intricate problem. A train passes a 270 metre platform in 38 seconds, and passes a man standing on that platform in 20 seconds. Find the train's speed and length.
Think about what happens. In 38 seconds, the train covers its own length plus 270 metres. In 20 seconds, it covers just its own length. The difference — 18 seconds — corresponds to 270 metres. So speed equals 270 divided by 18, which equals 15 metres per second, which is 54 kilometres per hour.
The train's length equals distance covered in 20 seconds: 15 times 20 equals 300 metres.
Let us recap the key takeaways from today's lesson.
First, the three fundamental formulas: speed equals distance over time, distance equals speed times time, and time equals distance over speed.
Second, always ensure your units match before calculating — convert when necessary.
Third, to convert kilometres per hour to metres per second, multiply by 5/18; to convert metres per second to kilometres per hour, multiply by 18/5.
Fourth, average speed equals total distance divided by total time — never simply average the speeds.
Fifth, relative speed is the sum of speeds for opposite directions, and the difference for same directions.
Sixth, for train problems, remember: pole means train length only, platform means train length plus platform length.
That brings us to the end of our lesson on Speed, Distance and Time. Practice these concepts with care, pay attention to your units, and work through problems step by step. You have all the tools you need to succeed. Keep up the excellent work, and I will see you in the next lesson.