I double my speed, why will I take half the time?

The most important relationship in this concept is:

·         Distance = Speed × Time

·         Speed = Distance ÷ Time

·         Time = Distance ÷ Speed

Note: The units used in the three quantities should either be in SI or MKS system.

Now let us discuss a problem:

If I double my speed, why will I take half the time?

The reason is that, speed and time has an inverse relationship, provided the distance remains constant.

See the following example to understand this concept:

·         If A travels 1500 m at 5 kmph, then the time taken
= 1500m ÷ 5kmph = 1.5 ÷ 5 = 0.3 hr.
= 0.3 × 60 min = 18 min.

·         If A doubles his speed, i.e. 10 kmph, then time taken
= 1.5 ÷ 10 hr = 0.15 hr.
= 0.15 × 60 min = 9 min, which is half of 18 min.

Note: If the distance is constant, then the ratio of speeds of the moving objects is reverse of the ratio of the time taken by those and vice versa.

Example 1:

The ratio of speeds of two men, A and B is 1:2. Find the ratio of time taken by them to travel 10 km each.

Solution:

We know that, for equal distance traveled, the ratio of their time taken is reverse of the ratio of their speeds.

So answer will be 2:1.

Example 2:

A runs a race of 1 km at a speed of 6 kmph and B at a speed of 12 kmph. How many meters start can B give A?

Solution:

A covers 1 km in 1/6 hrs = 10 min, and B in 1/12 hrs = 5 min.

Since, A takes 5 min longer than B, the start that B can give A is the distance traveled by A in 5 min which is one-half of a km i.e. 500 meters.

Or, we can simply use the fact that whenever speeds are in the ratio 1:2, the start that the faster one can give to the slower one in a d km race is always half of d.

Example 3:

A man travels a total distance of 5 km. After every one km he doubles his speed. If he covers 5 km in 15hrs and 30 min, find his speed during the first 1km.

Solution:

The speeds at which the man traveled are in the ratio, 1 : 2 : 4 : 8 : 16.
Thus, time taken to travel every km is in the ratio 16 : 8 : 4 : 2 : 1.
The total time taken to travel 5km = 15 hrs 30 min = 15.5 hrs.
Therefore, the time taken to travel the first 1 km = 16/31 × 15.5 hrs = 8 hrs. 

So, speed during the first one km = 1/8 kmph

Example 4:

A man travels for 1hr at a speed of 20 kmph and for the next one hour he doubles his speed. Find the average speed at which he traveled for two hours.

Solution:

Average speed = total distance/total time.
Total distance traveled in two hours = 20 + 40 = 60 km.

Therefore, average speed = 60/ = 30 kmph