In distance-speed relations, we will learn about the Distance formula. We will also see velocity formula. The problems that are from this section base themselves on the concepts and definitions of terms that measure distance, speed and time. Here we will see many examples from this section, we will introduce the important concepts like the Distance formula, the velocity formula etc. We will also solve questions based on the Distance formula and the other relevant formulae. Let us start!

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## Distance Formula – Distance/Speed Relation

First, we will collect all the formulae that we shall use to solve the examples ahead. Let us define the terms that will appear in the formulae. Let us say that an object starts moving from point A. The motion means that it changes its position with respect to some fixed point in its surroundings.

**Browse more Topics Under Time And Speed**

- Distance/Speed Relation
- Train Problems
- Data Sufficiency
- Relative Speed and Conversions
- Time & Speed Practice Questions

**Distance**

The distance is the actual length of the path that the object travels.

**Displacement**

The displacement is the shortest distance between the initial position and the final position of the journey. For example, if an object goes in a circle and returns to its initial position, then its displacement is zero while its distance is not. In general, we say that distance is either greater than or equal to the displacement of an object. The displacement can never be smaller than the distance.

**Velocity Formula **

The velocity is the time rate of change of displacement. If ‘S’ is the displacement of an object in some time ‘T’, then the velocity is equal to, v = S/T. The units of velocity are m/s or km/hr.

**Speed:**

The speed is the time rate of change of the distance. If ‘D’ is the distance of an object in some time ‘T’, the speed is equal to, s = D/T. It has the same units as velocity. let us solve some examples, we will introduce the formulae as we go along.

## Examples

Example 1: The speed of a bus is 54 km/h if we don’t let it stop at any point. If the bus stops at the bus-stops, the speed of the bus is 45 km/h. What is the time that the bus stops for per hour?

A) 7 min B) 10 min C) 21 min D) 22 min

Answer: The relation between distance-speed and time is, speed = Distance/Time. Therefore, we have

Distance = Speed × Time. As the speed of the bus is 54 km/h, it will cover 54 km in one hour. Similarly, when we let the bus stop, it will cover 45 km/h. Therefore, due to stoppages, it covers 9 km lesser.

Time taken to cover 9 km = [(Distance/Speed)] = [(9/54)] h = (9/54)×60 min = 10 min. Therefore the correct option is B) 10 min.

Example 2: Khan can cover a certain distance in 1hr 24min. He covers 2/3 of the distance at 4 km/h and the rest at 5 km/h. What is the total distance that Khan covers?

A) 3 km B) 4 km C) 5 km D) 6 km

Answer: Let the total distance be = D. Also the total time = 1hr and 24min = 1 (24/60) h or 84/60 h. Since we have that Khan covers 2/3 of the distance at 4 k/.h, we can write, for this part of the journey, speed = 4km/h and distance = [2/3]D.

For the rest of the journey, we have speed = 5 km/h and the distance Khan covers = D – [2/3]D = (1/3)D

Hence, from the formula for speed, we have: 84/60 hr = [{(2/3) D}/4] + [(1/3)D/5].

We will have to simplify this to find the value of D from it. We have:

(21/15) h = (14/3)D×3

or D = 6 km. Therefore the total distance that Khan covers = 6 km and option D) 6km is the correct choice.

### Some More Examples

Example 3: A certain motor car starts at the speed of 70 km/hr. It accelerates, increasing its speed every two hours by an amount of 10 km/h. What is the time it will take to cover a distance of 345 km?

A) 2 (1/4) hrs B) 4 hrs 5 min C) 4 (1/2) hrs D) Can not be determined

Answer: The distance that the car covers in the first 2 hours = (70 x 2) km = 140 km

Also, the distance that the car covers in the next 2 hours = (80 x 2) km = 160 km

Thus, remaining distance = 345 – (140 + 160) = 45 km.

Therefore the speed in the fifth hour = 90 km/hr. Also the time the car takes to cover 45 km = (45/90) hr = (1/2) hr

Therefore the total time taken = 2 + 2 + (1/2) = 4 (1/2) hrs. Hence the correct option is C) 4 (1/2) hours.

Example 4: A person ‘A’ can complete a journey in 19 hours. If he travels the first half of this journey at the rate of 21 km/hr and second half of the journey at the rate of 24 km/hr, then find the total length of the journey in km.

A) 214 km B) 224 km C) 234 km D) 244 km

Answer: Let the total distance of the journey be = x km. Then, from the distance formula, we have:

Time of the first half + Time of the second half = 19 h

[x/2]/21 + [x/2]/24 = 19. Here we have to find the value of the ‘x’ from the equation. Therefore, we have x = 224 km.

## Practice Questions

Q 1: A person travels from P to Q at a speed of 40 km/h and returns by increasing his speed by 50%. What is his average speed for both the trips?

A) 34 km/h B) 44 km/hr C) 48 km/hr D) 52 km/hr

Answer: C) 48 km/hr

Q 2: A man on a scooter moves at a certain speed. If he increases his speed by 3 km/h he would take 40 min lesser. If he moves 2km/h slower he would take 40 min more. The distance he travels is:

A) 20 km B) 30 km C) 40 km D) 50 km

Answer: C) 40 km