Distance, Speed and Time

# Distance, Speed and Time

## 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.
c. If a cyclist covers 60 m in 12 seconds, his speed is 5 m/sec.

Speed is a measure of how quickly an object moves from one place to another. It is equal to the distance traveled divided by the time. It is possible to find any of these three values using the other two. This picture is helpful:

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.

### 1.    km/h to m/s conversion

x km/h = (x x 5/18) m/s

### 2.    m/s to km/h conversion

 x m/s = (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