SPEED, DISTANCE AND TIME

Speed should be handled with almost care both in our daily life and in Exams too. In competitive exams we should attend the questions carefully with maximum speed so that these can be answered in minimum time, so that you can become a Government servant in minimum time.
*Speed of a moving body is the distance travelled in unit time.
Speed =Distance/Time

TYPE-I

1.If A travels a certain distance at xkm/hr and B travels the same distance at y km/ hr, then the average speed of the whole journey is
2xy x  y
2.If an object travels at a speed of x km/ hr for half of the distance and y km/hr for the remaining half, then the
average speed = 2xy/xy
3.If a man goes at a speed of x km/hr for a certain distance and then another distance at a speed of y km/hr,average 
speed is given by :xy/xy

EXAMPLES

1.Find the total distance travelled if a car travelled for 10 hours with a speed of 20 km/hr for half of the distance and with a speed of 40 km/hr for the remaining half distance.
Distance = time x average speed
=t*2xy /x  y
=10x2x20x40/2040
=10*40*40/60=266.66
.'.Distance = 266.67 km
2.Find the total distance travelled, if a bus travelled for 20 hours with a speed of 15 km/hr for certain distance and with a speed of 10 km/hr the remaining distance.
Distance = time x average speed
=t*2xy/xy
=20 x10x15/10  15
=20x10x15/25= 120
.'.Total distance = 120 km

TYPE - II

If a man travelling at a speed of'x km/ hr' is late by'a minutes' and if he travels at a speed of 'y km/hr' reaches earlier by'b minutes', then the distance of travel is given by:
D=product of Speeds/Difference of Speeds x ^(Difference in time) 
ie, D=xy/x-y*(a-b)

EXAMPLES

1.A man walking with a speed of 10 m/s reaches the office 5 minutes late. If he walks at a speed of 15 m/s, he will reach the office 5 minutes earlier. Find the distance to office.
D=(xy/x-y)(a-b)
ie,D=(15*10/15-10)*10
Here (a - b) = 10 min. (5 min earlier and 5 min. late)
= 150/5*10 = 300 m 
.'.Distance = 300m
2.Two bike racers covered a distance at a speed of 60 km/hr and 50 km/hr respectively by a difference of 3 minutes. How much distance did they cover.
D=(xy/x-y)(a-b)
=60*50/60-50*3
=60x50x3/10=900
.'.Distance = 900km

TRAINS

We all travel in trains as ticket rates are lower than buses and makes our journey more comfortable. In the same way problems related to trains are also very easy to study and makes you happy.
If two bodies are in motion and their speeds are a and b respectively, then
(a)Relative speed = ab , if they are ; moving in opposite directions
(b)Relative speed = a-b , if they are  moving in the same direction.

TYPE -I

*If a train of length 'x metres' passes a post it travels a distance equal to its length.
ie,x metres
*If a train of length 'x1 metres' passes a bridge or a platform of length 'x2 metres', the running train will travel a distance
of '(x1  x2 )’metres

EXAMPLES

1.A train 100m long travels at 50 km/hr. In what time will it pass a man who is , walking at 5 km/hr in the
(a)same direction
(b)opposite direction 
(a) Relative speed = 50 - 5 = 45 km/hr 
= 45X5/18=25/2 M/S
Required time=100/25/2 = 8 seconds
(b) Relative speed = 50  5 =55 km/hr
= 55 x5/18=15.27 m/s 18
Required time=100/15.27
= 6.54 seconds
2.A train travels 90 km/hr. How many metres will be travelled in 20 minutes.
Speed of the train =[90x5/18]= 25 m/s
Distance covered= 20 x 60 sec 
= 25 x 20 x 60 
= 30,000 m

TYPE -II

*If a train of length x1m and speed v1 m/ s and another train of length x2m and speed v2 m/s are running on two parallel tracks in the same direction, then the faster train will overtake the slower train
in x1x2/v1-v2 seconds if v1 > v2 and
x1 x2/ v2 > V1 seconds  if v2 > v1 
*If the trains are moving in the opposite directions, they will pass each other in
Xl X2/v1v2 seconds

EXAMPLES

1.

Two trains of the same length but with different speeds pass an electirc pole in 6 seconds and 7 seconds respectively. In what time will they cross each other when they are moving in
(a)the same direction
(b)opposite direction
Let the length of the train be x metres
Speed of the first train = x/6m/s 
Speed of the second train = x/7m/s
Total distance travelled = x  x = 2x metres
(a) Relative speed =x/6-x/7=x/42m/s
.'.Required time =2x/x/42=84 seconds
(b) Relative speed = x/6x/7=13x/42
.'.Required time =2x/13x/42=84/13 
=6.46 seconds

2.

A train A leaves Trivandrum at 5 am and reaches Ernakulam at 9 am. Another train leaves Ernakulam at 7 am and  reaches Trivandrum at 10.30 am. At; what time do the two trains cross each other.
Let the distance between Trivandrum and Ernakulam be x km and let the trains meet y hrs after 7 am
Given, A covers x km in 4 hrs and
B covers x km in 7/2 hrs
.'. Speed of A = x/4 km/hr
Speed of B=2x/7 km/hr
Distance covered by A in (y2)hrs
Distance covered by y hrs=x
.'.x/4(y2)2x/7*y=x
ie,y2/4 , 2y/7=1
y=14/15 hrs
=14/15*60 min 
= 56 min
.'.The trains meet at 7.56 am

NOTE

If two trains start at the same time from points A and B towards each other and after crossing they take a and b sec. in running B and A respectively, then
A's speed : B's speed = /b :/a

PRACTICE EXERCISES 

1.A train running at the speed of 54 km/hr crosses a pole in 10 seconds. What is the length of the train.
2.Two trains running is opposite directions cross a man standing on the platform in 27 seconds and 17 seconds respectively and they cross each other in 23 seconds. The ratio of their speeds is____
3.Two trains,one from Kollam to palakkad and the other from palakkad to kollam started simultaneously.After they met,the trains reached their destinations after 16 hours and 25 hours respectively.Find the ratio of their speeds
4.Two trains are moving in opposite directions at the speed of 50 km/hr and 80 km/hr. Their lengths are 2 km and 1 km respectively. The time taken by the slower train to cross the faster train in seconds is 
5.An athlete practices running at 9 km/hr alongside a railway track in 240 metres ahead of the engine of a 120 metre long train running at 45 km/hr in the same direction. In how much time will the train pass the athlete.
6.A train 600 metres long is running at a speed of 72 km/hr. If it crosses a tunnel in 2 minutes, then the length of the tunnel in metres is ____
7.A train travels at the rate of 60 km/hr without stoppages and it travels at 40 km/ hr with stoppages. How many minutes does the train stop on average per hour.
8.The length of the bridge, which a train 160m long, travelling at 36 km/hr can cross in 50 seconds is ____ 
9.How many seconds a 1000 metre long train take to cross a man walking with a speed of 2 km/hr in the direction of the moving train if the speed of the train is 74 km/hr.
10.Two trains 200m and 180m long run at the speed of 36 km/hr and 54 km/hr respectively in opposite directions on
parallel tracks.The time which they take to cross each other is __

ANSWERS WITH EXPLANATIONS

1.

Speed = 54 km/hr = 54x5/18m/s = 15 m/s
Time = 10s
.'.Length of the train = 15 x 10 = 150m

2.

Let the speeds of the two trains be x m/ s and y m/s respectively.
Then,
Length of the first train = 27 x metres
Length of the second train = 17y metres
.'.27x17y/xy = 23
ie, 23x  23y = 27x  17y
4x =6y
x/y =3/2
.'.Required ratio =3:2

3.

A's speed : B's speed = /b:/a
=/25:/16=5:4 
.'.Ratio of speeds = 5:4

4.

Relative speed = (5080) = 130 km/hr
= 130x5/18m/s 
=(325/9)m/s
Distance covered = 2  1 = 3km = 3000m
Required time = 3000x9/325= 27000/325 = 83.07 second

5.

Speed of train relative to athlete 
= (45 - 9) km/hr 
= 36 km/hr
= 36x5/18 =10 m/s
Distance to be covered
= (240  120)m 
= 360m
.'.Time taken = 360/10 sec. =36 sec.

6.

Speed = (72x5/18) = 20 m/s
Time = 2 min = 2 x 60 = 120 sec.
Let the length of the tunnel be x metres
Then,600x/120=20
600x=2400
x =1800m
.'.Length of the tunnel is 1800 m.

7.

Due to stoppages, it covers 20km [(60 - 40)km] less per hour.
Time taken to cover 20km = 20/60*60
= 20 min.
Hence, the train stops on an average 20 min.

8.

Speed =(36*5/18)m/s=10m/s
Time=50s
Let the length of the bridge be x metres
Then,160  x/50 = 10
160  x = 500
X = 340
.'.Length of the bridge = 340m

9.

Speed of the train relative to man 
=(74-2)
=(72X5/18)m/s = 20 m/S
.'.Time taken to pass the man
1000/20
=50 seconds

10.

Relative speed = 36  54 = 90. km/hr
= (90 x5/18) m/s
= 25 m/s
Distance = 200  180 = 380m
Time = Distance/Speed = 380/25
= 15.2 seconds

BOATS AND STREAMS

Have you ever thought of questions related to boats and streams while you are enjoying your travel in boats. These questions can be easily answered by studying formulae.

DOWNSTREAM & UPSTREM

In water, the direction along the stream is called Downstream and the direction against the stream is called Upstream.
*If the speed downstream is x km/hr and the speed upstream is y km/hr,then
Speed in still water = 1/2(x  y )km / hr 
Rate of stream =1/2(x -y )km / hr
*If the speed of a boat in still water be x km/hr and that of stream be y km/hr, then
Speed of the boat downstream = (x  y ) km / hr 
Speed of the boat upstream = (x - y )km / hr

EXAMPLES

1.

A man swims at 8 km/hr in the direction of the stream and at 4 km/hr against the stream. What is his speed in still water.
Let the speed of the man in still water be x km/hr and that of stream be y km/hr.
.'. x  y = 8 and x - y = 4
x  y = 8
x-y = 4
2x = 12
x = 4
Hence, the speed of the man in still water is 4 km/hr

2.

A boat rows a distance of 40 km in 8 hours down stream and returns to the same point in 5 hours. Find the speed of boat in Stillwater.
Speed of the boat in the 
direction of the stream 
=40/8 = 5 km/hr 
Speed of the boat 
against the stream 
=40/5= 8km/hr
Let the speed of the boat in still water be x km/hr and speed of the stream be y km/hr.
Then, x  y = 5 
x-y = 8 
.'.2x = 13 
x = 6.5,
y=1.5
.'.The speed of boat in Stillwater = 6.5km/hr

3.

A person rows a distance of 34km in the direction of the stream in 6 hours and a distance of 27km in 6 hours against the stream. What is the speed of the stream.
Speed of the person in the 
direction of the stream
=34/6km/hr
Speed of the person 
against the stream 
=27/6km/hr
Let the speed of the person in still water be x km/hr and the speed of the stream be y km/hr.
Then, x  y = 34/6 ______(i)
x-y=27/6_________(ii)
Solving (i) and (ii)=>
2y=7/6
y=7/12=0.58
.'.Speed of the stream = 0.58 km/hr