Time and Distance – Formula, Units, Conversions, Relationship between Time Speed and Distance

# Time and Distance – Formula, Units, Conversions, Relationship between Time Speed and Distance

Time Speed and Distance is a major concept in Mathematics. Time and Distance are used extensively for questions relating to topics like circular motion, boats, and streams, motion in a straight line, clocks, races, etc. This article gives you a complete idea of the Relationship Between Time Speed and Distance, Units, Conversions, etc. Get Formula for Time and Distance, Solved Examples explaining the concept in detail.

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## Time Speed and Distance – Definition

Speed is a concept in motion that is all about how slow or fast an object travels. Speed is defined as the distance divided by Time. Speed, distance, and time are given to solve for one of the three variables when a piece of certain information is known.

### Relationship between Time Speed and Distance

Speed = Distance/Time

Speed tells us how fast or slow an object travels and describes the distance traveled divided by the time taken to cover the distance.

From the above formula, Speed is directly proportional to Distance and inversely proportional to Time.

### Units of Time Speed and Distance

Speed Distance and Time has different units for each of them and they are given as under

Time: seconds(s), minutes (min), hours (hr)

Distance: (meters (m), kilometers (km), miles, feet

Speed: m/s, km/hr

If Distance and Time Units are known Speed Units can be derived easily.

### Time Speed and Distance Conversions

• To change between km/hour to m/sec, we multiply by 5/18. So, 1 km/hour = 5/18 m/sec
• To change between m/sec to km / hour, we multiply by 18/5. So, 1m /sec = 18/5 km/hour = 3.6 km/hour
• Similarly, 1 km/hr = 5/8 miles/hour
• 1 yard = 3 feet
• 1 mile= 1.609 kilometer
• 1 kilometer= 1000 meters = 0.6214 mile
• 1 mile = 1760 yards
• 1 mile = 5280 feet
• 1 hour= 60 minutes= 60*60 seconds= 3600 seconds
• 1 yard = 3 feet
• 1 mph = (1 x 5280) / (1 x 3600) = 22/15 ft/sec
• 1 mph = (1 x 1760) / (1 x 3600) = 22/45 yards/sec
• For a certain distance, if the ratio of speeds is a : b, then the ratio of times taken to cover the distance is given by b : a and vice versa.

### Applications of Time Speed and Distance

Check out different models of questions asked on the concept of Time Speed and Distance. They are as under

#### Average Speed

Average Speed = Total Distance Travelled/Total Time Taken

Case 1: When Distance is Constant, Average Speed is given by = 2xy/x+y where x, y are two speeds at which the same distance is covered.

Case 2: When Time taken is Constant, Average Speed = (x+y)/2 where x, y are two speeds at which we traveled for the same time.

Examples

1. A person travels from one place to another at 40 km/hr and returns at 160 km/hr. If the total time taken is 5 hours, then find the Distance?

Solution:

Here the Distance is constant, so the Time taken will be inversely proportional to the Speed.

Ratio of Speed = 40:160

= 1:4

Ratio of Time = 4:1

Time taken = 5 hours

Therefore, time taken while going is 4 hours and while returning is 1 hour

Distance = Speed* Time

= 40*4

= 160 Km

Therefore, Distance Travelled is 160 Km.

2. Traveling at 3/2 nd of the original Speed a train is 20 minutes late. Find the usual Time taken by the train to complete the journey?

Solution:

Let the usual Speed be S1 and the usual Time is T1. As the Distance to be covered in both cases is the same,

The ratio of usual Time to the Time taken when he is late will be the inverse of the usual Speed and the Speed when he is late

If the Speed is S2 = S1 then the Time taken T2 = 3/2 T1 Given T2 – T1 = 20 =>3/2 T1 – T1 = 20 => T1 = 40 minutes.

#### Inverse Proportionality of Speed & Time

Speed is inversely proportional to time when the distance is constant. If Speeds are in the ratio of m:n then the time taken will be n:m They are two methods of solving questions.

Using Inverse Proportionality
Using Constant Product Rule

Example

After traveling 60km, a train is meeting with an accident and travels at (2/3)rd of the usual Speed and reaches 30 min late. Had the accident happened 15km further on it would have reached 20 min late. Find the usual Speed?

Solution:

Using Inverse Proportionality

Here there are 2 cases

Case 1: accident happens at 60 km

Case 2: accident happens at 75 km

The difference between the two cases is only for the 15 km between 60 and 75. The time difference of 10 minutes is only due to these 15 km.

In case 1, 15 km between 60 and 75 is covered at (2/3)rd Speed.

In case 2, 15 km between 60 and 75 is covered at the usual Speed.

So the usual Time “t” taken to cover 15 km, can be found out as below. 3/2 t – t = 10 mins = > t = 20 mins, d = 15 km

so Usual Speed = 15km/20min = 15km/0.33hr= 45Km/hr

Using Constant Product Rule Method

Let the actual Time taken be T

There is a (1/3)rd decrease in Speed, this will result in a (1/2)nd increase in Time taken as Speed and Time are inversely proportional

The delay due to this decrease is 10 minutes

Thus 1/2 T= 10 and T=20 minutes

Also, Distance = 15 km

Thus Speed = 15/20 minutes = 15km/0.33hr = 45km/hr

#### Meeting Point

If two people travel from points A and B towards each other they meet at point P. Distance covered by them on the meeting is AB. Time taken by both to meet is the same. Since Time is constant Distance AP and BP will be in the ratio of their Speeds.

Consider the distance between A and B is d.

If two people walking towards each other from A and B. when they meet for the first time they cover a distance of “d ” and when they meet for the second time they cover a distance of “3d” and when they meet for the third time they cover a distance of “5d” …..

Example

1. Amar and Arun have to travel from Delhi to Jaipur in their respective cars. Arun is driving at 45 kmph while Amar is driving at 60 kmph. Find the Time taken by Amar to reach Jaipur if Arun takes 6 hrs?

Solution:

Since the Distance covered is constant in both cases, the Time taken will be inversely proportional to the Speed.

From the given data, the Speed of Amar and Arun are in ratio 45:60 or 3:4.

So the ratio of the Time taken by Arun to that taken by Amar will be in the ratio 4:3. So if Arun takes 6 hrs, Amar will take 4.5 hrs.

### Solved Examples on Time and Distance

1. Seetha is driving a car with a speed of 60 km/hr for 1.5hr. How much distance does she travel?

Solution:

Speed = 60 Km/hr

Time = 1.5 hr

Distance = Speed *Time

= 60*1.5

= 90 Km

Therefore, Sheela Travels a distance of 90km.

2. While going to an office, Ram travels at a speed of 35 kmph, and on his way back, he travels at a speed of 40 kmph. What is his average speed for the whole journey?

Solution:

In this case, Distance is constant

Average Speed = 2xy/x+y where x, y is the speeds at which the distance is covered

Substitute the Speeds from the given data

Average Speed = (2*35*40)/35+40

= 37.33 km/hr

The Average Speed of the Whole Journey is 37.33Kmph

3. Ramu and Somesh are standing at two ends of a room with a width of 40 m. They start walking towards each other along the width of the room with a Speed of 4 m/s and 3 m/s respectively. Find the total distance traveled by Ramu when he meets Somesh for the third time?

Solution:

When Ramu meets Somesh for the third time he would have covered a distance of 5d i.e. 5*40m = 200m

The ratio of Speed of Ramu and Somesh is 4:3 so the Distance traveled by both of them will also be in the ratio of 4:3

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