ICSE Class 8 Maths Arithmetical Problems Chapter 05 Speed Time Distance

Speed, Time and Distance

The speed of a body is defined as the distance covered by it in unit time.

Speed = \(\frac{\text{distance}}{\text{time}}\), distance = speed \(\times\) time, time = \(\frac{\text{distance}}{\text{speed}}\)

Usually, when the distance is measured in kilometres (km), the unit of time is the hour (h) and when the distance is measured in metres (m), the unit of time is the second (s).

Examples

(i) If a car covers 40 km in 30 min, it would cover 80 km in 1 h. So, its speed is 80 km/h.

(ii) If a swimmer covers a distance of 200 m in 2 min, his speed is 200 m - 120 s = 1.7 m/s (nearly).

Table 5.1 Corresponding Units Of Distance, Time And Speed

Teacher's Note

Speed is everywhere in daily life - from checking how fast you run to understanding vehicle speeds on highways.

Conversion Of Units

(i) Let us convert km/h into m/s and cm/s.

1 km/h = \(\frac{1000 \text{ m}}{60 \times 60 \text{ s}}\) = \(\frac{1000}{3600}\) m/s = \(\frac{5}{18}\) m/s = \(\frac{5}{18} \times 100\) cm/s = \(\frac{250}{9}\) cm/s.

1 km/h = \(\frac{5}{18}\) m/s = \(\frac{250}{9}\) cm/s

(ii) Let us convert m/s into km/h.

\(\frac{5}{18}\) m/s = 1 km/h.

1 m/s = \(\frac{18}{5}\) km/h

(iii) Let us convert cm/s into km/h.

\(\frac{250}{9}\) cm/s = 1 km/h. Therefore, 1 cm/s = \(\frac{9}{250}\) km/h.

Teacher's Note

Understanding unit conversions helps when reading speed limits in different countries that use different measurement systems.

Worked Example

A man walks 100 m in 3 min.

(i) Find his speed in km/h.

(ii) What distance would he cover in 2 h 15 min?

(iii) How long would he take to cover 3.5 km?

Solution

(i) Speed = \(\frac{\text{distance}}{\text{time}}\) = \(\frac{100 \text{ m}}{3 \text{ min}}\) = \(\frac{100}{3 \times 60}\) m/s = \(\frac{5}{9}\) m/s = \(\frac{5}{9} \times \frac{18}{5}\) km/h = 2 km/h.

(ii) The distance he would cover in 2 h 15 min (i.e., \(\frac{9}{4}\) h)

= speed \(\times\) time = (2 km/h) \(\times\) \(\left(\frac{9}{4} \text{ h}\right)\) = 4.5 km.

(iii) The time he would take to cover 3.5 km

= \(\frac{\text{distance}}{\text{speed}}\) = \(\frac{3.5 \text{ km}}{2 \text{ km/h}}\) = \(\frac{3.5}{2}\) h = 1 h 45 min.

Teacher's Note

This example shows how a simple walking pace of 2 km/h is a typical human walking speed - helpful for planning travel times on foot.

Uniform Speed

In the preceding example we assumed that the man walks at exactly the same pace all the time. In other words, we assumed that he covers equal distances in equal intervals of time. When a person or a body does this, we say that he/she/it has uniform speed.

For example, if a car covers a distance of 16 m in a second, its speed will be uniform provided it covers 16 m in each subsequent second.

Teacher's Note

Real-world vehicles rarely maintain perfectly uniform speed - they speed up and slow down due to traffic and road conditions.

Average Speed

In reality, however, things and people mostly do not have uniform speed. They pick up speed and slow down during the course of their travel. So, we calculate the average speed by dividing the total distance covered by a body by the total time it takes to cover the distance.

Average speed = \(\frac{\text{total distance covered}}{\text{total time taken}}\)

Worked Example

A train covers the first 30 km of its journey at the speed of 60 km/h and the next 60 km at the speed of 80 km/h. Find its average speed.

Solution

Time taken to cover 30 km at the speed of 60 km/h = \(\frac{\text{distance}}{\text{speed}}\) = \(\frac{30}{60}\) h = \(\frac{1}{2}\) h.

Time taken to cover 60 km at the speed of 80 km/h = \(\frac{60}{80}\) h = \(\frac{3}{4}\) h.

Total distance covered = 30 km + 60 km = 90 km

and total time taken = \(\frac{1}{2}\) h + \(\frac{3}{4}\) h = \(\frac{5}{4}\) h.

Hence, average speed of the train = \(\frac{\text{total distance covered}}{\text{total time taken}}\) = \(\frac{90}{\frac{5}{4}}\) km/h = 72 km/h.

Teacher's Note

Average speed on a long journey is useful for estimating arrival times - a 360 km journey at an average of 72 km/h takes 5 hours.

Some Important Points

1. When a moving train crosses a pole (or a point or a person who is stationary), it covers a distance equal to its own length.

Therefore, time taken to cross the pole = time taken to cover a distance equal to the length of the train = \(\frac{\text{the length of the train}}{\text{the speed of the train}}\)

2. When a moving train crosses a platform (or a bridge), it covers a distance equal to the sum of its length and the length of the platform.

Therefore, time taken to cross a platform = \(\frac{\text{the length of the train} + \text{the length of the platform}}{\text{the speed of the train}}\)

Teacher's Note

These concepts explain why longer trains take more time to completely pass through a station or cross a bridge.

Worked Example

The length of a train is 140 m and its speed is 72 km/h. How long will it take to cross (i) a pole and (ii) a platform 220 m long?

Solution

The speed of the train = 72 km/h = 72 \(\times\) \(\frac{5}{18}\) m/s = 20 m/s.

(i) The distance travelled by the train in crossing the pole = the length of the train = 140 m.

Therefore, the time taken to cross the pole = \(\frac{\text{distance travelled}}{\text{speed}}\) = \(\frac{140}{20}\) s = 7 s.

(ii) Here, the distance travelled by the train to cross the platform = the length of the train + the length of the platform = 140 m + 220 m = 360 m.

Therefore, time taken to cross the platform = \(\frac{\text{distance travelled to cross the platform}}{\text{speed of the train}}\)

= \(\frac{360}{20}\) s = 18 s.

Teacher's Note

This explains the difference in time - crossing a 140 m train takes 7 seconds, but crossing it plus a 220 m platform takes 18 seconds.

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