Time and Distance: Formulas, Shortcuts and Solved Examples

What is speed?

Speed is the distance covered in one unit of time. So Speed = Distance / Time.

A bus covers 180 km in 6 hours. Its speed is 180 / 6 = 30 km/hr.

Sources: NCERT Class 7 Science: Motion and Time, measuring speed; Practice question: speed of a bus from distance and time.

What are the three forms of the formula?

Any one of the three can be found from the other two.

• Speed = Distance / Time
• Distance = Speed × Time
• Time = Distance / Speed

Sources: NCERT Class 7 Science: Motion and Time; Practice question: time to run 400 metres.

How do you convert between km/hr and m/sec?

One km is 1000 m. One hour is 3600 seconds. So 1 km/hr is 1000/3600 = 5/18 m/sec.

To go from km/hr to m/sec, multiply by 5/18. To go from m/sec to km/hr, multiply by 18/5.

Sources: NCERT Class 7 Science: Motion and Time, measuring speed; Practice question: 18 km/hr in m/sec.

What happens when the distance is fixed?

For a fixed distance, a higher speed gives a lower time. Speed and time are in inverse ratio.

If two cars have speeds in the ratio 2 : 9, the times for the same distance are in the ratio 9 : 2.

Sources: NCERT Class 7 Science: Motion and Time; Practice question: time ratio from speed ratio.

How do you find average speed?

Average speed is total distance divided by total time. Do not average the speeds.

Suppose the distances are equal, with speeds x and y. Then the average speed is 2xy / (x + y).

Sources: NCERT Class 7 Science: Motion and Time, average speed; Practice question: average speed for 4 and 28 km/hr.

What if a person walks at a fraction of the usual speed?

At a/b of the usual speed, the time becomes b/a of the usual time.

So the extra time is (b − a)/a of the usual time. Set it equal to the delay and solve.

Sources: Practice question: walking at 2/3 of the usual speed.

How do you use two speeds with early and late arrival?

The distance is the same in both cases. So d/x − d/y equals the total gap in time between the two arrivals.

If one speed is 20 minutes late and the other is 40 minutes early, the gap is 60 minutes.

Sources: Practice question: distance from late and early arrival.

What is relative speed?

When two objects move towards each other, their speeds add. When they move in the same direction, the faster one gains by the difference.

Then divide the gap between them by the relative speed to get the time.

Sources: NCERT Class 7 Science: Motion and Time.

Example 1: Convert 54 km/hr to m/sec and 25 m/sec to km/hr.

Step 1. 54 × 5/18 = 3 × 5 = 15 m/sec.

Step 2. 25 × 18/5 = 5 × 18 = 90 km/hr.

Why this works. One km/hr is 5/18 of a m/sec. So the factors are exact inverses of each other.

Example 2: How far does a car go in 2 hours 40 minutes at 45 km/hr?

Step 1. Convert the time: 40 minutes = 2/3 hour. So the time is 2 2/3 = 8/3 hours.

Step 2. Distance = 45 × 8/3 = 120 km.

Why this works. The unit of time must match the unit of speed. Here both are in hours.

Example 3: A man goes 60 km at 30 km/hr and comes back at 60 km/hr. What is his average speed?

Step 1. Time going = 60/30 = 2 hours. Time returning = 60/60 = 1 hour.

Step 2. Total distance = 120 km. Total time = 3 hours.

Step 3. Average speed = 120/3 = 40 km/hr.

Check. 2 × 30 × 60 / (30 + 60) = 3600/90 = 40. The shortcut agrees.

Why this works. He spends more time at the slower speed. So the average is closer to 30 than to 45.

Example 4: By walking at 3/4 of his usual speed, a man reaches 20 minutes late. What is his usual time?

Step 1. New time = 4/3 of the usual time T.

Step 2. Extra time = 4T/3 − T = T/3. This equals 20 minutes.

Step 3. T = 20 × 3 = 60 minutes.

Check. New time = 4/3 × 60 = 80 minutes. That is 20 more than 60.

Why this works. A slower speed means a longer time. The fraction flips for time.

Example 5: Towns A and B are 180 km apart. A traveller leaves A at 40 km/hr and another leaves B at 50 km/hr towards A. When do they meet?

Step 1. Relative speed = 40 + 50 = 90 km/hr.

Step 2. Time = 180/90 = 2 hours.

Step 3. A has covered 40 × 2 = 80 km. B has covered 50 × 2 = 100 km. Together that is 180 km.

Why this works. Each hour, the gap shrinks by both distances. So we add the speeds.

What quick check catches big mistakes?

Check the units first. Then check the size. A walking speed of 90 km/hr is clearly wrong.

For average speed, the answer must lie between the two speeds. In Example 3, 40 lies between 30 and 60.