Speed, Distance and Time

1.1 What is speed?

Speed is the total distance covered divided by the total time taken. The textbook uses the word speed to mean average speed.

Speed = Total distance ÷ Total time

The idea is old. NCERT says it goes back to the Aryabhatiya, a text of the 5th century CE.

1.2 How do I find distance or time?

Rearrange the same formula. Distance covered equals speed times time. Time taken equals distance divided by speed.

Our own example: a bus runs at 60 km/h for 2.5 hours. It covers 60 × 2.5 = 150 km. Run it backwards: 150 km at 60 km/h takes 150 ÷ 60 = 2.5 hours.

1.3 Which units do I use?

The basic unit of speed is metre per second, written m/s. Kilometre per hour, written km/h, is also common. A speedometer shows speed directly in km/h.

Write unit symbols in the singular: 50 km, not 50 kms.

• Why it matters: the formula only works when distance and time use matching units.
• Example: NCERT says rockets reach orbit at up to 8 km/s, while a tortoise moves at about 8 cm/s.
• Misconception: the same number does not mean the same speed. 8 km/s and 8 cm/s are very different.

2.1 What is uniform motion?

Uniform motion is travel along a straight line at a constant speed. The object covers equal distances in equal intervals of time. Its average speed then equals its actual speed.

2.2 What is average speed?

Average speed is the total distance travelled divided by the time taken to cover it. Real trips rarely keep one speed. Motion is called non-uniform when the speed keeps changing.

2.3 Can I just average two speeds?

No. A simple average treats both halves of a trip as if they took equal time. They do not.

Our own example: a car drives 120 km at 60 km/h, then returns 120 km at 40 km/h. The first leg takes 2 hours. The second takes 3 hours.

Total distance is 240 km and total time is 5 hours. Average speed is 240 ÷ 5 = 48 km/h. The simple average, 50 km/h, is wrong.

• Why it matters: slower legs last longer, so they pull the average down.
• Example: 60 km/h out and 40 km/h back gives 48 km/h, not 50 km/h.
• Misconception: the average of two speeds is not the average speed of the trip.

2.4 Does speed have a direction?

No. Speed tells us how fast an object moves but gives no information about its direction. Average speed has only a number.

Velocity adds a direction. That matters in the relative-speed cards below.

3.1 How do I change km/h to m/s?

Multiply by 5/18. One kilometre is 1,000 m and one hour is 3,600 s. So 1,000 ÷ 3,600 reduces to 5/18 (our own arithmetic).

Example: 72 km/h × 5/18 = 20 m/s. To go back from m/s to km/h, multiply by 18/5. So 10 m/s × 18/5 = 36 km/h.

3.2 Is there a quick table?

Each row is the km/h value times 5/18. Multiples of 18 convert cleanly.

3.3 What is the usual unit trap?

Mixing minutes with hours, or metres with kilometres. Convert everything to one unit first.

Our own example: 45 minutes at 40 km/h. Change 45 minutes to 0.75 hour. Then distance is 40 × 0.75 = 30 km. Using 45 as hours would give 1,800 km.

4.1 What is relative speed?

Relative speed tells how fast one object moves as seen from another. In one dimension, the velocities add. OpenStax gives a train moving east at 10 m/s.

A person walks toward the back of the train at 2 m/s relative to the train. Adding the two gives 8 m/s east relative to the Earth.

4.2 What if two objects move toward each other?

Add their speeds. NCERT’s postmen start 210 yojanas apart and walk toward each other. One walks 9 yojanas per day and the other 5.

Together they cover 9 + 5 = 14 yojanas a day. So they meet after 210 ÷ 14 = 15 days. The first covers 135 yojanas and the second 75.

4.3 What if they move the same way?

Subtract the speeds. The faster one gains on the slower one only by the difference.

Our own example: a car at 60 km/h chases a bike at 40 km/h, 100 km ahead. The gap closes at 20 km/h. The car needs 100 ÷ 20 = 5 hours.

• Why it matters: the gap shrinks only by the difference of speeds.
• Example: CAPF 2026 asks how far A must run to catch B (see 6.2).
• Misconception: the catching-up time is not the gap divided by the faster speed.

4.4 How long does a train take to cross a pole or a platform?

It must cover its own length to clear a pole. It must cover its length plus the platform’s length to clear a platform. This is our own reading of the problem.

Example: a 150 m train at 54 km/h moves at 15 m/s. It clears a pole in 150 ÷ 15 = 10 s. It clears a 150 m platform in 300 ÷ 15 = 20 s.

5.1 How far away is the lightning?

Multiply the delay by the speed of sound. Light reaches you almost at once, so the delay is the sound’s travel time (our own reading).

Example: thunder comes 6 seconds after the flash. At 320 m/s the distance is 320 × 6 = 1,920 m.

5.2 Why must I read the clock times carefully?

Questions may give clock times, not a delay. Subtract the earlier time from the later one first.

• Why it matters: the formula needs the gap, not the two readings.
• Example: 19:06:23 minus 19:06:17 is 6 seconds.
• Misconception: an echo travels out and back, so its distance is halved. Thunder travels one way, so it is not.

Read the echo idea in Sound Waves and Acoustics.

6.1 What do examiners test here?

They test small slips, not hard sums. Five traps repeat.

• Unit mismatch. Minutes with hours, or metres with kilometres.
• Averaging speeds. Use total distance over total time.
• Add or subtract. Toward each other adds. The same way subtracts.
• Length of train. Add it to the distance to be crossed.
• Clock times. Subtract them before using the formula.

6.2 What have exams asked?

CAPF stands for Central Armed Police Forces. ACs stands for Assistant Commandants. Each label links to that paper.