Time and Distance

1. BASIC CONCEPT

Time and Distance problems deal with the relationship among distance, time and speed. These questions are common in competitive examinations and can be solved quickly by understanding the basic formula and unit conversion.

Fundamental Formula:
Speed = Distance ÷ Time
Distance = Speed × Time
Time = Distance ÷ Speed

Three important quantities

Speed: Distance travelled per unit of time.
Distance: Total length travelled.
Time: Duration required to cover a distance.

2. UNITS AND CONVERSION

Speed is generally expressed in km/h or m/s.

1 km = 1000 m
1 hour = 60 minutes = 3600 seconds

``` 1 km/h = 5/18 m/s
1 m/s = 18/5 km/h ```

Example 1: Convert 72 km/h into m/s.

72 × 5/18 = 20 m/s.

Example 2: Convert 15 m/s into km/h.

15 × 18/5 = 54 km/h.

3. DIRECT APPLICATION OF FORMULA

Example 3: A car travels 180 km in 4 hours. Find its speed.

Speed = 180 ÷ 4 = 45 km/h.

Example 4: A cyclist moves at 12 km/h for 3 hours. Find the distance travelled.

Distance = 12 × 3 = 36 km.

Example 5: A bus travels at 60 km/h and covers 240 km. Find the time taken.

Time = 240 ÷ 60 = 4 hours.

4. SPEED-TIME RELATION

For a fixed distance:
Speed ∝ 1/Time

Therefore, if speed increases, time decreases, and if speed decreases, time increases.

Example 6: A journey takes 8 hours at 40 km/h. At 64 km/h, the same journey takes:

New Time = 8 × 40/64 = 5 hours.

Quick Rule: For the same distance, S₁T₁ = S₂T₂.

5. AVERAGE SPEED

Average speed is not always the simple average of two speeds. It depends on the distances or times involved.

Average Speed = Total Distance ÷ Total Time

When Equal Distances Are Covered

Average Speed = 2ab/(a+b)

Example 1: A person travels equal distances at 40 km/h and 60 km/h. Find average speed.

Average Speed = (2 × 40 × 60)/(40 + 60)
= 4800/100 = 48 km/h.

When Equal Times Are Involved

Average Speed = (a+b)/2

Example 2: A vehicle moves at 30 km/h for 2 hours and 50 km/h for 2 hours.

Average Speed = (30 + 50)/2 = 40 km/h.

6. RELATIVE SPEED

Relative speed is the speed of one object with respect to another.

Objects moving in the SAME direction:
Relative Speed = Difference of speeds = |a − b|

``` Objects moving in OPPOSITE directions:
Relative Speed = Sum of speeds = a + b ```

Example 3: Two cars move in the same direction at 70 km/h and 50 km/h.
Relative Speed = 70 − 50 = 20 km/h.

Example 4: Two cars approach each other at 60 km/h and 40 km/h.
Relative Speed = 60 + 40 = 100 km/h.

7. MEETING TIME

Meeting Time = Distance between objects ÷ Relative Speed

Example 5: Two persons are 150 km apart and travel towards each other at 40 km/h and 35 km/h.

Relative Speed = 40 + 35 = 75 km/h
Time = 150/75 = 2 hours.

8. SPEED RATIO AND TIME RATIO

For the same distance:
Speed Ratio = a:b
Time Ratio = b:a

Example 6: A and B cover the same distance. Their speeds are in the ratio 3:4.

Their times will be in the ratio 4:3.

Remember: Speed and time are inversely proportional when distance remains constant.

9. PERCENTAGE CHANGE IN SPEED

If speed increases by x%, time decreases by:
x/(100+x) × 100%

``` If speed decreases by x%, time increases by:
x/(100−x) × 100% ```

Example 7: Speed is increased by 25%. Percentage reduction in time:

25/125 × 100 = 20%.

10. TRAINS

Train problems are based mainly on relative speed and the distance that the train must cover.

Train crossing a pole/person:
Time = Length of train ÷ Speed of train

``` Train crossing a platform:
Time = (Length of train + Length of platform) ÷ Speed

Two trains crossing each other:
Time = Sum of lengths ÷ Relative speed ```

Example 1: A train 180 m long runs at 54 km/h. Find the time required to cross a pole.

54 km/h = 54 × 5/18 = 15 m/s
Time = 180/15 = 12 seconds.

Example 2: A train 150 m long crosses a 250 m platform at 72 km/h.

Total distance = 150 + 250 = 400 m
72 km/h = 20 m/s
Time = 400/20 = 20 seconds.

11. TWO TRAINS

Example 3: Two trains of lengths 200 m and 300 m move in opposite directions at 54 km/h and 36 km/h.

Relative speed = 54 + 36 = 90 km/h = 25 m/s
Total length = 500 m
Time = 500/25 = 20 seconds.

Example 4: Two trains move in the same direction at 72 km/h and 54 km/h. Their lengths are 180 m and 120 m.

Relative speed = 18 km/h = 5 m/s
Total length = 300 m
Time = 300/5 = 60 seconds.

12. BOATS AND STREAMS

In boat and stream problems, the speed of the boat changes according to the direction of the stream.

Downstream Speed = Boat speed + Stream speed
Upstream Speed = Boat speed − Stream speed

``` Boat speed in still water = (Downstream Speed + Upstream Speed) ÷ 2

Stream speed = (Downstream Speed − Upstream Speed) ÷ 2 ```

Example 5: A boat moves downstream at 18 km/h and upstream at 10 km/h. Find the speed of the boat in still water and the stream.

Boat speed = (18 + 10)/2 = 14 km/h
Stream speed = (18 − 10)/2 = 4 km/h.

13. ROUND TRIP

Example 6: A boat covers 30 km downstream at 15 km/h and returns upstream at 10 km/h.

Downstream time = 30/15 = 2 h
Upstream time = 30/10 = 3 h
Total time = 5 hours
Total distance = 60 km
Average speed = 60/5 = 12 km/h.

Important: For equal distances at speeds a and b, average speed is 2ab/(a+b).

14. RELATIVE MOTION — QUICK TABLE

Situation	Relative Speed
Same direction	\|a − b\|
Opposite direction	a + b
Boat downstream	b + s
Boat upstream	b − s

15. COMPLETE FORMULA SHEET

Topic	Formula
Speed	Distance ÷ Time
Distance	Speed × Time
Time	Distance ÷ Speed
km/h → m/s	× 5/18
m/s → km/h	× 18/5
Equal distance average	2ab/(a+b)
Same direction	Relative speed = \|a−b\|
Opposite direction	Relative speed = a+b
Train & pole	L/S
Train & platform	(L+P)/S
Downstream	B+S
Upstream	B−S

16. SOLVED REVISION EXAMPLES

Example 1: A man covers 24 km in 3 hours. Find his speed.
Speed = 24/3 = 8 km/h.

Example 2: A car travels at 80 km/h for 2.5 hours. Find distance.
Distance = 80 × 2.5 = 200 km.

Example 3: Two people are 210 km apart and move towards each other at 60 km/h and 45 km/h.
Relative speed = 105 km/h
Time = 210/105 = 2 hours.

Example 4: A train 240 m long crosses a pole in 12 seconds. Find speed.
Speed = 240/12 = 20 m/s
= 20 × 18/5 = 72 km/h.

Example 5: A person travels half of a journey at 30 km/h and the other half at 50 km/h.
Average speed = (2 × 30 × 50)/(30 + 50)
= 3000/80 = 37.5 km/h.

17. PRACTICE QUESTIONS

A car covers 360 km in 6 hours. Find its speed.
Convert 90 km/h into m/s.
A man walks at 5 km/h for 4 hours. Find the distance.
A journey of 300 km is covered at 60 km/h. Find the time.
Two cars travel in the same direction at 75 km/h and 55 km/h. Find relative speed.
Two persons 180 km apart move towards each other at 40 km/h and 50 km/h. Find meeting time.
Find the average speed for equal distances travelled at 30 km/h and 60 km/h.
A train 210 m long runs at 63 km/h. Find the time to cross a pole.
A boat's downstream speed is 20 km/h and upstream speed is 12 km/h. Find boat speed and stream speed.
A vehicle's speed is increased by 20%. Find the percentage decrease in time for the same distance.

18. ANSWERS