## Time and Distance – Formulas, Questions and Solved Examples

** **

The speed of a moving body is defined as ** the distance traversed by it in unit time.** Thus, when we say that the speed of a car is 40km/hr, we mean that the car traverses 40km in 1 hour. If a body moving at a speed of

**V m/s**travels a distance of

**S mt**in

**t seconds.**

Then, **V = S/t **or** S = Vt **or** t = S/V.**

**FORMULAS:**

**SPEED = DISTANCE/ TIME****DISTANCE = SPEED X TIME****TIME = DISTANCE/SPEED****AVERAGE SPEED = TOTAL DISTANCE/ TOTAL TIME TAKEN**

**RELATION BETWEEN THE UNITS OF DISTANCE COVERED, SPEED AND TIME ARE:**

SPEED |
DISTANCE |
TIME |

Km/hr |
Km (kilometer) | hr (hour) |

m/min |
m(meter) | min (minute) |

m/sec |
m(meter) | sec (second) |

Cm/sec |
cm (centimeter) | sec (second) |

CONVERSION

- To convert a speed from km/hr into a speed in m/s, we must multiply it by
**5/18** - Conversely, to convert a speed from m/s into km/hr we must multiply it by
**18/5**

**Let’s discuss some problems on time, speed and distance.**

*1.Gini is going on her moped at aspeed of 24km/hr. how much time will she take to cover 540 metre?*

** Solution: ** Conversion: speed = 24km/hr = (24 X5)/18 m/sec = 20/3 m/sec

By unitary method,

20/3 m distance is covered in 1 second

1 m distance is covered in 1/(20/3)sec

540 m distance is covered in (1 X 540)/ (20/3) = 81 sec

*Therefore, gini takes 1 min 21 sec to cover 540 m.*

*2.A train in first two hours travels 108 km, in next 30 minutes covers 3 km and in next two hours covers 90 km . What is the average speed of whole journey travelled by the train?*

*2.A train in first two hours travels 108 km, in next 30 minutes covers 3 km and in next two hours covers 90 km . What is the average speed of whole journey travelled by the train?*

** Solution: ** Total distance covered = (108+36+90) km = 234 km.

** **Total time taken = (2+1/2+2) = 9/2 hours

Average speed = __total distance covered__

Total time taken

= (234)/(9/2) km/hr

= 26 X 2 = 52 km /hr

*Therefore, the average speed of the train during the whole journey was 52km/hr.*

* *

*3.The distance between two towns is covered in 5 hours at the speed of 60 km/hr.If the speed is increased by 15 km/hr then what amount of time will be saved? *

*3.The distance between two towns is covered in 5 hours at the speed of 60 km/hr.If the speed is increased by 15 km/hr then what amount of time will be saved?*

** Solution: ** Distance between the two towns = speed X time = (60 X 5) = 300 km

In second case, the speed should be increased by 15 km /hr to save time. So new speed will be

= (60 + 15) = 75 km/hr.

The time taken = distance/ speed

= 300/75 = 4 hrs.

*Therefore, the time saved is (5-4) = 1 hour.*

* *

*4.A distance of 450 km is covered in 6*^{ 1}/_{2} If the first 2/3 rd of the distance is covered at the speed of 75 km/hr, at what speed the remaining distance should be covered?

*4.A distance of 450 km is covered in 6*

^{ 1}/_{2}If the first 2/3 rd of the distance is covered at the speed of 75 km/hr, at what speed the remaining distance should be covered?** Solution: ** 2/3 rd of the whole distance = (2/3) X 450 = 300 km.

And the speed with which this distance is covered= 75 km/hr

The time taken to cover this distance = __Distance__

Speed

= 300/75hrs = 4 hrs.

Remaining distance = ( 450 – 300) = 150 km.

150 km distance to be covered and time taken is = (6^{1}/_{2}– 4) ={ (13/2) – 4} = 5/2 hrs= 2.5 hrs.

The required speed is = (150/2.5)= 0 km/hr.

*Therefore, the required speed will be 60 km/hr.*

* *

*5.The first 300 km of a journey is covered in 3 *^{1 }/_{3 }hrs and remaining of the journey is covered in 2^{2}/_{3} hrs at the speed 75km/hrs. find the average speed of the whole journey.

*5.The first 300 km of a journey is covered in 3*

^{1 }/_{3 }hrs and remaining of the journey is covered in 2^{2}/_{3}hrs at the speed 75km/hrs. find the average speed of the whole journey.* *

** Solution: **In the second part of journey the distance covered = Speed X Time

** = **75 X (2

^{2 }/

_{3}) = (75 X 8/3) = 200 km.

In the first part of the journey distance is 300 km.

So, the total distance is = (200 + 300) = 500 km

The total time = 3^{1 }/_{3}+ 2^{1 }/_{3} = ( 10/3 + 8/3) = 18/3 = 6 hrs.

The average speed = __Total distance covered__

Total time taken

= 500/6 = 83^{1}/_{3} km/hrs

*Therefore, the average speed of the whole journey is = 83 ^{1 }/_{3}km/hrs.*

* *

*6.A truck covers a distance of 550m in 1 min, whereas a bus covers a distance of 33 km in ¾ hr. calculate the ratio of their speed.*

*6.A truck covers a distance of 550m in 1 min, whereas a bus covers a distance of 33 km in ¾ hr. calculate the ratio of their speed.*

* *

** Solution: **1 min = 60 sec

So, speed of the truck = (550/60) = 55/6 m/s

Speed of the bus in km/hr = (33) /3/4

= (33 X 4)/3 = 44 km/hr

Speed of the bus in m/s = (44 X 5) /18 = 110/9 m/s

Ratio of

Speed of truck : Speed of bus :: 55/6 : 110/9 = 3/4

*Therefore, the ratio of their speed is 3 : 4.*

* *

RELATIVE SPEED (1) |
Difference of the speed when two bodies are moving in same direction. |

RELATIVE SPEED (2) |
Sum of the speed, when the two bodies are moving in opposite direction. |

* *

*7.Two trains are running at the rate of 40km/hr and 50km/hr respectively on parallel rail line in opposite direction. If they take 10s to pass each other completely and one of the train is 120m long, then find the length of the other train.*

*7.Two trains are running at the rate of 40km/hr and 50km/hr respectively on parallel rail line in opposite direction. If they take 10s to pass each other completely and one of the train is 120m long, then find the length of the other train.*

* *

** Solution: **The speed of one train relative to the other train = ( 40 + 50) = 90 km/hr

* = *__90 X 5 __m/s = 25m/s

** **18

Distance travelled in 10 second = 25 X 10 = 250m.

So, sum of the length of the two trains = 250 m

Length of one train is = 120m

So, length of another train = (250 – 120 ) = 130 m

*Therefore, length of the other train is 130 m.*

* *

*8.A train is travelling at the speed of 40 km/hr leaves Delhi at 8:00 am and another train travelling at the speed of 60 km/hr leaves Delhi at 10:30 am in the same direction. Find the distance from Delhi when two trains willm meet?*

*8.A train is travelling at the speed of 40 km/hr leaves Delhi at 8:00 am and another train travelling at the speed of 60 km/hr leaves Delhi at 10:30 am in the same direction. Find the distance from Delhi when two trains willm meet?*

** Solution: ** Difference between the time of starting of two trains = ( 10:30 – 8:00)= 2 hrs 30 min = 5/2 hrs.

So, before starting second train , the distance covered by the first train in 5/2 hrs = 40 X 5/2 = 100 km.

In every hour, the gain of distance by second train over first train = (60 – 40) = 20 km/hr

Second train will cover the distance (100/20) = in 5 hrs.

In 5 hrs the distance from Delhi = (60 X 5) = 300 km.

* Therefore, both the trains will be together at 300 km from Delhi.*

*IN CASE OF PROBLEMS IN boats and stream WE HAVE TO ALWAYS IMAGINE *

* In still water the boat is having speed = x km/hr.*

*Speed of the stream = y km/hr*

So, speed of boat in the direction of current or downstream = **(x + y) km/hr**

Speed of the boat in the direction opposite to the current or upstream = **(x – y) km/hr**

**HERE ARE SOME PROBLEMS ON STREAM and BOATS**

*9.The speed of a boat in still water is 7km/hr. if a boat is rowed upstream for a distance of 22km in 4 hours, find the speed of the current. *

*9.The speed of a boat in still water is 7km/hr. if a boat is rowed upstream for a distance of 22km in 4 hours, find the speed of the current.*

* *

** Solution : ** Let the current speed or stream speed =

**x km/hr**

Speed of boat in upstream = (**7-x) km/hr**

In 4 hr the boat is rowed to reach 22 km

In 1 hr the boat is rowed to reach 22/4 km.

According to the question,

(7-x)= 22/4

Or, 4(7-x) = 22

28-4x = 22

4x = 28-22

Or, x= 6/4= 3/2 km/hr

*Therefore, the speed of the stream or current = 1.5km/hr.*

*Therefore, the speed of the stream or current = 1.5km/hr.*

* *

*10.The speed of the current in river is 2km/hr. if a boat is rowed downstream for a distance of 34.5 km in 3 hours, then find the speed of boat in still water.*

*10.The speed of the current in river is 2km/hr. if a boat is rowed downstream for a distance of 34.5 km in 3 hours, then find the speed of boat in still water.*

** Solution: **Let in the still water the boat speed be =

**x km/hr**

** **Speed of the boat in downstream =** (x+2) km/hr**

In 3 hrs the boat is rowed = 34.5 km

In 1 hour the boat is rowed = 34.5/3 = 11.5 km

According to the question,

x+2 = 11.5

Or, x = 11.5 – 2

Or, x = 9.5

*Therefore, the speed of the boat in still water is 9.5 km/hr.*