Distance, Speed and Time
The
speed of a moving object is the distance covered by it in a unit time.
For
example,
a. If
the speed of a car is 60 km/h, it means it covers 60 km in 1 hour.
b. If a car covers 150 km in 2 hours, its speed is 75 km/h.
b. If a car covers 150 km in 2 hours, its speed is 75 km/h.
c. If a cyclist covers 60 m in 12
seconds, his speed is 5 m/sec.
The
positions of the words in the triangle show where they need to go in the
equations. To find the speed, distance is over time in the triangle, so speed
is distance divided by time. To find distance, speed is beside time, so
distance is speed multiplied by time.
Distance, Speed and Time Formula
Speed = Distance/Time, Time = Distance/Speed, Distance = Speed x Time
Units of Speed
Speed is
normally expressed in km/h (kilometer per hour) or m/sec (meter per second). We
can convert the units from km/h into m/sec or vice versa.
Conversion of units of speed
1. km/h to m/s conversion
x km/h = (x x 5/18) m/s
2. m/s to km/h conversion
(x x
|
18/5) km/h
|
Uniform Speed, Variable Speed and Average Speed
When an
object covers equal distances in equal intervals of time, its speed is said to
be
uniform, otherwise the speed is said to be variable. In practice, a vehicle does not
cover
equal
distances at a uniform speed. The speed goes on varying because of various
reasons. In such cases, we determine the total distance covered and divide it
by the total time to get the speed, which is called the average speed.
Average speed = Total distance covered/Total time
taken
Example 1: An express train moves with a speed
of 144 km/h. Calculate:
a. its
speed in m/sec.
b. the
distance covered by the train in 15 minutes?
c. the
time taken by the train to cover 56 km?
Solution:
a. We
know that km/h is converted to m/sec by multiplying the given speed in km/h by
5/18.
144 km/h
= 144 × 5/18 m/sec = 40 m/sec
b.
Distance covered = Speed × Time = 144 × ¼ km (15 minutes = ¼ h) = 36 km
c. Time
= Distance covered/Speed = 56 km/144 km/h = 7/18 hour = 7 × 60/18 minutes = 70/3
minutes = 23 minutes 20 seconds
Example 2: A car covers the first 250 km of a
journey in 3⅓ hours and the remaining of the journey at a speed of 60 km/hr in
2⅔ hours. Find the average speed of the car for the whole journey.
Solution: For the first journey, 250 km is
covered in 3⅓ hours.
For the
second journey, distance covered = speed × time = 60 × 8/3 km = 160 km
Total
distance covered = (250 + 160) km = 410 km
Total
time taken = (3⅓ hours + 2⅔ hours) = 6 hours
Thus, average speed = Total distance covered/Total time taken
= 410 km/6
h = 68⅓ km/h
Problems on Trains
1. When
a train is passing through a pole or a signal pole or a standing person, the
train
travels
a distance equal to its length.
2. When
a train is passing through a platform, a tunnel or a bridge, the train travels
a distance equal to the sum of the
length of the train and the length of the object (platform, tunnel or a
bridge).
Example 3: A train is running at a speed of 72
km/h. It crosses a man in 12 seconds. Find the length of the train.
Solution: Speed of the train = 72 km/h = (72 ×
5/18) m/sec = 20 m/sec
Time
taken to cross the man = 12 sec
Length
of the train = distance covered = speed × time = 20 m/sec × 12 sec = 240 m
Thus,
length of the train is 240 m.
Example 4: A 260 m long train crosses a bridge
190 m long in 45 sec. Find the speed of the train in km/h.
Solution: Distance covered by train = 260 m +
190 m = 450 m
Time
taken = 45 sec
Speed of
the train = Distance covered/Time taken = 450 m/45 sec = 10 m/sec = 10 × 18/5 km/h = 36 km/h
Thus,
the speed of the train is 36 km/h.
Relative Speed
While
sitting in a train and observing out of the window, have you experience the
following.
1. When
our train is crossing another train coming from the opposite direction, it
appears that the other train is coming with the higher speed.
2. When
another train is moving along or in same direction with our train, it appears
that the trains are moving at a very slow speed.
Relative
speed between two moving objects means the speed of one object with respect to the
other.
1. When
two objects are moving in the opposite directions, then their relative speed is
the sum of their speeds.
If two
trains A and B are running in the opposite direction with speed 40 km/h and 60
km/h respectively, then after one hour, the trains will be 100 km apart. Thus,
the relative speed is (40 + 60) km/h.
2. When
two objects are moving in the same direction, then their relative speed is the
difference
of their speeds.
If two
trains A and B are moving in the same direction with speed 40 km/h and 60 km/h
respectively, then after one hour, the trains will be only 20 km apart. Thus,
the relative
speed is
(60 – 40) km/h.
Example 5: Two trains of lengths 300 m and 330
m are running at a speed of 72 km/h
and 54
km/h respectively on parallel tracks. How long will it take to pass each other,
if
they run
in
a. the
same direction? b. the opposite
directions?
Solution: Distance covered to pass each other
= 300 m + 330 m = 630 m (length of both trains)
a. When
the trains are running in the same direction, the relative speed will be difference
of their speeds, i.e., (72 – 54) km/h = 18 km/h = 18 × 5/18 m/sec = 5 m/sec.
Time
taken to cross each other = distance/speed = 630/5 = 126 sec = 2 minutes 6 seconds
b. When
the trains are running in the opposite direction, the relative speed will be
sum of their speeds, i.e., (72 + 54) km/h = 126 km/h = 126 × 5/18 m/sec = 35
m/sec.
Time
taken of cross each other = distance/speed = 630/35 m/sec = 18 seconds.
Example 6: A train running at a speed of 60
km/h leaves Delhi at 10.00 a.m. and another
train
running at a speed of 75 km/h leaves Delhi at 12 noon in the same direction.
How far from Delhi will the two trains are together?
Solution: Difference between the start or
departure time of the two trains = (12 – 10) = 2 hours
Distance
covered by first train in 2 hours = 60 km × 2 = 120 km
The
relative speed of the two trains = 75 km/h – 60 km/h = 15 km/h
The
second train will gain 120 km over the first train in 120/15 hours, i.e., 8 hours
Thus,
the required distance from Delhi = 75 km/h × 8 hours = 600 km.
Example 7: When two trains are running in the
opposite direction at a speed of 40 km/h and 32 km/h respectively, the faster
train passes a man sitting in the slower train in 15 seconds. Find the length
of the faster train.
Solution: The relative speed of the trains =
(40 + 32) km/h = 72 km/h = 72 × 5/18 m/sec
= 20
m/sec
Here, the
faster train passes a man sitting in the slower train in 15 seconds.
Length
of the faster train = distance covered in 15 seconds = 20 × 15 meters = 300 meters.
Hence,
the length of the faster train is 300 meters.
Problems on Streams
While
dealing with problems based on streams, find the relative speed of boats/swimmers
as given below.
Let the
speed of a boat (or a swimmer) in still water be x km/h and the speed of the stream be y km/h.
The
speed of the boat will be slower than its speed in still water when it moves
upstream (against the stream). Thus, the relative speed of the boat upstream
will be (x – y) km/h.
The
speed of the boat will be faster than its speed in still water when it moves
downstream (along the stream). Thus, the relative speed of the boat downstream
will be (x + y) km/h.
Example 8: A man can swim 5 km/h in still
water. The speed of the stream is 3 km/h.
How much
time will the man take to swim 4 km if
a. he is
swimming along the stream?
b. he is
swimming against the stream?
Solution: Speed of the man in still water = 5
km/h
Speed of
the stream = 3 km/h
Speed of
the man along the stream = (5 + 3) km/h = 8 km/h
Speed of
the man against the stream = (5 – 3) km/h = 2 km/h
a. Time
taken to swim 4 km along the stream = distance/speed = 4/8 hour = ½ hour
b. Time
taken to swim 4 km against the stream = distance/speed = 4/2 hours = 2 hours
Example 9: The speed of a boat in still water
is 20 km/h. If the boat goes upstream for a
distance
of 20 km in 4 hours, find the speed of the stream.
Solution: Let the speed of the stream be x km/h.
The boat
covers a distance of 20 km in 4 hours upstream.
Speed of
the boat upstream = distance/time = 20/4 = 5 km/h
Speed of
the boat in still water = 20 km/h
Speed of
boat upstream = 20 – x
5 = 20 –
x
x = 20 – 5 = 15
Speed of
the stream = 15 km/h
Example 10: The speed of a boat in still water
is 10 km/h and the speed of the stream is
2 km/h.
Find
a. the
time taken by the boat to go 72 km downstream.
b. the
time taken by the boat to go 32 km upstream.
Solution:
a. Speed
of the boat downstream = (10 + 2) km/h = 12 km/h
Distance
traveled downstream = 72 km
Time
taken by the boat downstream = distance/speed = 72/12 = 6 hours
b. Speed
of the boat upstream = (10 – 2) km/h = 8 km/h
Distance
traveled upstream = 32 km
Time
taken by the boat upstream = distance/speed = 32/8 = 4 hours