Time and Distance - Notes | Study Quantitative Techniques for CLAT - CLAT

Speed is a very basic concept in motion which is all about how fast or slow any object moves. We define speed as distance divided by time.

Distance is directly proportional to Velocity when time is constant.

Few Formulae

• Distance travelled = speed x time
• Speed = distance / tim
• Time = distance / speed1
  km/hr = (5/18) meter / sec
  1 metre / sec = 18 / 5 km / hr.    

If the time taken, to travel two distances with different speeds, is equal then

Average speed = (speed₁ + speed₂) / 2

Also     

• average speed = total distance travelled  / total time taken  
• if two distances travelled are equal and two speeds are x km /hr and y km / hr then average speed =
• if 3 equal distances are travelled at speeds x km /hr, ykm/ hr and z km/hr respectively then average speed =

 Formulae for Questions on Train

• To pass a pole or signal post or a standing man, a train has to travel a distance equal to its length. So, the time taken to pass a pole = length of the train/ speed off the train 
• To pass a platform of length A, the train has to travel a distance equal to its length + the length of the platform (A). So, time taken by the train (length T) to pass a platform (length A) = T+A / Speed
• Let the speeds of two trains be x km / hr and y km /hr then
  The relative speed if the direction of the trains is the same = (x – y) km/ hr
  The relative speed if the direction of trains is opposite to each other
  = (x+y) km /hr

Example 1: A bike crosses a bridge at a speed of 180 km/hr. what will be length of the bridge if the bike takes 3 minutes to cross the bridge. 

Solution: 
Length of the bridge = distance travelled by the bike in 3 minutes = speed x time

Speed = 180 km/hr = (180 x 5) / 18 = 50 meters / sec.

Time = 3 x 60 = 180 seconds.

So, length of the bridge

= 50 X 180 meters

= (50 x 180) / 1000 km

= 9 km

Example 2: Two persons are moving in the direction opposite to each other. The speeds of the two persons are 8 km/hr and 5 km per hour. Find their relative speed wrt each other.

Solution: We know that when two objects move in the direction opposite to each other, the relative speed is the sum of two speeds.

So, required relative speed = 8 + 5 = 13 km/hr

Example 3: Two trains A and B are moving at the speeds in the ratio of 2 : 5. Find the ratio of the time taken to travel the same distance. 

Solution: The ratio of time taken is inverse of the speeds of the two cars.

i.e. since speeds are 2 : 5

so, time must be 5 : 2

Example 4: Prem can cover a certain distance in 42 minutes by covering 2/3 of the distance at 4 km/hr and the rest at 5 km/hr. find the total distance. 

Solution: Let total distance be x

So, as per the given condition.

Distance / speed + distance / speed = total time

Or (2x/3) x ¼ + (x/3) x (1/5) = 42/60

Or x/2 + x/5 = 42/20

7x/10 x= 42/20

X = 3 km

Example 5: A man completes 60 km of a journey at 12 km / hr and the remaining 80 km of the journey in 5 hours. Find the average speed for the whole journey. 

Solution: Total distance travelled = 60 km + 80 km = 140 km

Total time taken = 5 hours + 5 hours = 10 hours

So, average speed for the whole journey = 140/10

= 14 km/hr

Speed during the first part = 12 km/hr

Time taken during the first part to cover 60 km = 60 / 12= 5 hours

Time taken to cover the second part = 5 hrs

And the speed during 2^nd part = 80/5 = 16 km/hr

So, the time taken during the two journeys is equal

So, average speed = (12 km/hr + 16 km/hr) / 2 = 28/2

= 14 km/hr

Example 6: A car travels from A to B at a speed of 58 km/hr and travels back from B to A at the speed of 42 km/hr. what is average speed of the car in covering the distance both ways. 

Solution: Since the car travels equal distance with different speed, the average speed is the harmonic mean of the two speeds.

Average speed =

Example 7: The speed of A and B are in the ratio 3 : 4. A takes 20 minutes more than the time taken by B to reach a destination. In what time doesA reach the destination

Solution: Let time taken by A be x hours.

Then time taken by B = (x – 20/60) hours

Or (x – 1/3) hours

Ratio of speeds = inverse ratio of time

or x = 4/3 hours

Example 8: A train crosses a platform in 20 seconds but a man standing on the platform in 8 seconds. Length of the platform is 180 meters. Find the length of the train and its speed.

Solution: Time taken to cross the platform = 20 sec.

Time taken to cross the man = 8 sec.

20 – 8 = 12 sec is the time taken by the train to travel the distance equal to platform.

The train travels 180 meter in 12 seconds

So, its speed is 180/12 = 15 m/sec.

And the length of the train = distance travelled in

8 sec. = 15 x 8 = 120 meter

Example 9: A train running at 54 km/hr. takes 25 seconds to pass a platform and 15 seconds to pass a man walking at 6 km/h in the same direction in which train is going. Find the length of the train and the length of the platform. 

Solution: Let x and y be the length of the train and of the platform respectively.

Speed of the train w.r.t the man = 54 – 6 = 48 km/ph

= 48 x 5/18 = 40/3 meter / seconds

In passing the man the train covers its own length with the relative speed.

So the length of the train = Relative speed x time

= (40/3 x 15) m = 200 m.

Speed of the train = 54 x 5/18 = 15 m/second.

Distance travelled to pass the platform = x + y

So, x + y = 200 + y

So, (200 + y) / 15 = 25

Or y + 200 = 375

Or y = 175 meters.