# Speed, Time and Distance

The terms "Time" and "Distance" are related to the speed of a moving object. The questions on speed, time and distance are based on one general formula: **Distance = Speed × Time**

**Speed:** Speed of an object is the distance covered by it in a unit time interval. It is obtained by dividing the distance covered by the object, by the time it takes to cover that distance. For example: Suppose you travel from office (A) to home (B). The distance between A and B is 30 km. You travel at the speed of 40 km/hr and it takes 45 minutes.

### Concept

- If the time taken is constant, the distance travelled is proportional to the speed, that is, more the speed; more the distance travelled in the same time.
- If the speed is constant, the distance travelled is proportional to the time taken, that is, more the distance travelled; more the time taken at the same speed.
- If the distance travelled is constant, the speed is inversely proportional to the time taken, that is, more the speed; less the time taken for the same distance travelled.
- There are three parameters: Speed, Time and Distance. Keeping one parameter constant and changing another, the third parameter also gets changed.

**Units of Measurement**

Generally, if the distance is measured in kilometre, we measure time in hrs and speed in kilometre per hour and is written as km/h. If the distance is measured in metre then time is taken in second and speed in metre per second and is written as m/s.

1 km/hr = 1000 m /(60 × 60) s = 5/18 m/s

1 m/s = 18/5 km/h

**Example 1:** How long does a train 100 m long running at the rate of 40 Km/h take to cross a telegraphic pole?

In crossing the pole, the train must travel its own length.

Distance travelled is 100 m.

Speed = 40 Km/h = 40 × 5/18 m/s = 100/9 m/s

Time taken to cross the pole = 100/(100/9) = 9 seconds.

**Example 2:** A train running at a speed of 90 Km/h passes a pole on the platform in 20 s. Find the length of the train in metres?

Speed of the train = 90 Km/h = 90 × 5/18 m/s = 25 m/s.

Length of the train = Speed of the train × time taken in crossing the pole

= 25 × 20 = 500 m.

**Example 3:** A ship sails to a certain city at the speed of 15 knots/hr and sails back to the same point at the rate of 30 knots/hr. What is the average speed for the whole journey?

Let distance between the point and city d knots.

Total distance travelled = 2d

Total time taken = d/15 + d/30 = d/10

Average speed = Total distance / Total time

Average speed = 20 knots/hr

**Example 4:** A boy goes to school with the speed of 3 km an hour and returns with a speed of 2 km/h. If he takes 5 hrs in all, find the distance in km between the village and the school?

Let the distance between A and B be d km.

Time taken during onward journey = t_{1} = d/3

Time taken during return journey = t_{2} = d/2

Total time taken = 5 = d/3 + d/2 = 5d/6

d = 6 km.

**Example 5:** Nikita starts her journey from Delhi to Bhopal and simultaneously Nishita starts from Bhopal to Delhi. After crossing each other they finish their remaining journey in 49/9 hrs and 9 hrs, respectively. What is Nishita’s speed if Nikita’s speed is 36 Km/h?

Let Nishita’s speed be s km/hr.

Let distance between Delhi (D) and Bhopal (B) be DB. They meet at point P.

DP/36 = BP/s

After they meet at point P.

BP/36 = 49/9

BP = 196 km

DP/s = 9; DP = 9s

So,

9s/36 = 196/s

s^{2} = 196 × 4

s = 28 km/hr

**Example 6:** A car during its journey travels 40 minutes at a speed of 30 km/h, another 50 minutes at a speed of 60 km/h and 1 hr at a speed of 30 km/h. Find the average speed of the car?

Total time = 2/3 + 5/6 + 1 = 15/6 hours

Total distance = (2/3 × 30) + (5/6 × 60) + (1 × 30) = 100 km

Average speed = 100/(15/6) = 40 km/hr

**Example 7:** By walking at four-fifths of his usual speed, Mohan is 6 minutes late to his office. Find his usual time to cover the distance?

Let usual time be t minutes. Let speed be s. Distance travelled is same in both the cases.

4s/5 × (t + 6) = st

4t + 24 = 5t

t = 24 minutes.

**Example 8:** Without stoppages, a train travels certain distance with an average speed of 80 Km/h and with stoppages, it covers the same distance with an average speed of 60 Km/h. How many minutes per hour the train stops?

Let the distance travelled be d km.

Time taken by the train without stopping = d/80

Time taken by the train with stoppages = d/60

Total stoppage time = d/60 - d/80 = d/240

Stoppage time per hour = (d/240)/(d/60) = 1/4 hr

= 15 minutes

### Questions on Trains

A train has a definite length. The distance covered by the train depends on the length of the train.

If a train overtakes a pole or a man or a milestone, then the distance covered in overtaking is equal to the length of the train.

If a train overtakes a bridge or a tunnel or a platform or another train, then the distance covered is equal to the sum of the two lengths.

**Relative Speed**

If two trains of length L_{1} and L_{2} are moving in opposite directions at s_{1} and s_{2} speed, respectively, then then s_{1} + s_{2} is their relative speed.

If two trains of length L_{1} and L_{2} are moving in same directions at s_{1} and s_{2} speed, respectively, then then s_{1} - s_{2} is their relative speed.

Total distance travelled = L_{1} + L_{2}

Time taken to cross = Total distance / Relative speed

**Example 9:** A 600 m long train crosses a pole in 9 seconds. What is the speed of the train in Km/h?

Speed of the train = Length of the train / time taken in crossing the pole

= 600/9 m/s

= 600/9 × 18/5 km/h

= 240 km/hr

**Example 10:** A train 130 m long passes a bridge in 21 seconds moving with a speed of 90 Km/h. Find the length of the bridge?

Speed of the train = (Length of train + Length of bridge) / Time taken in crossing the bridge

90 × 5/18 = d/21

d = 525 m

Length of bridge = 525 - 130 = 295 m

**Example 11:** A train 135 m long is running with a speed of 49 Km/h. In what time will it pass a man who is walking at 5 Km/h in the direction opposite to that of the train?

Distance = 135 m

Relative speed = (49 + 5) km/h = 54 km/h

= 54 × 5/18 m/s = 15 m/s

Time = 135/15 = 9 s

**Example 12:** Two trains of length 110 metres and 90 m are running on parallel lines in the same direction with a speed of 35 km/h and 40 km/h, respectively. In what time will they pass each other?

Here, distance = 110 m + 90 m = 200 m

Relative speed = 40 km/h - 35 km/h = 5 km/h = 25/18 m/s

Time = 200/(25/18) = 144 s

**Example 13:** A train starts from Mumbai at 10 a.m. with a speed of 25 Km/h and another train starts from there on the same day at 3 p.m. in the same direction with a speed of 35 Km/h. Find at what distance from Mumbai both the trains will meet and find also the time of their meeting?

Distance travelled by both the trains is equal.

Extra time required by the first train = 3 pm = 10 am = 5 hours

So, 25 × (t + 5) = 35 × t

25t + 125 = 35t

t = 12.5 hours

They will meet at 3 pm + 12.5 hours = 3 a.m.

Distance = 35 × 12.5 = 437.5 km

**Example 14:** Two trains start at the same time from Delhi and Rohtak and proceed towards each other at the rate of 75 km/h and 65 km/h, respectively. When they meet, it is found that one train has travelled 10 km more than the other. Find the distance between Delhi and Rohtak?

Here, time travelled is same. If distance travelled by slower train is d km, then distance travelled by faster train is (d+10) km

So,

d/65 = (d + 10)/75

75d = 65d + 650

d = 65 km

Distance between Delhi and Rohtak = Combined distance travelled by both trains

= d + d + 10 = 140 km