### Key Concepts

1. Distance and displacement

2. Speed and velocity

3. Acceleration and its types

## Introduction

The motion of a body can be described using a few physical quantities, namely distance, displacement, speed, velocity, acceleration etc. In other words, the motion of a body can be analyzed by knowing those physical quantities. By knowing the present state of motion of a body, a prediction about its future motion can also be done.

### Explanation:

**Distance and Displacement **

The figure below shows two locations A and B which are connected by many paths, namely 1, 2, 3, 4, and 5. Each of these paths is of certain length. A body taking a certain path would need to cover the distance, which is equal to the length of the path taken by the body. Therefore, the body has to travel a larger distance if it chooses to take a longer path. Here, the path number 5 connects the locations A and B directly. Hence, it is the shortest path of all. The shortest distance between two points is called the **displacement**. Thus, if a body takes the path number 5, it covers the least possible distance to reach B from A.

**Displacement/ distance** covered by a body is measured in the units of kilometers, miles, foot, meters, etc. However, the SI unit of measurement is **‘meter’ (m)**.

**Problems and Solutions **

- Calculate the distance and displacement of the car in all the following cases by referring to the figure below and figure out whether the distance is equal to the displacement.

- The car moves from point P to S via Q and R and then again comes back to R.

- The car moves from point P to S via Q and R and then again comes back to Q.

- The car moves up to the mark of 65 km.

**Solution: **

- Distance = PQ + QR + RS + SR

= 40 + 45 + 25 + 25

= **135 km**

Displacement = PR = **85 km **

Distance and displacement are **not equal**.

- Distance = PQ + QR + RS + SR + RQ

= 40 + 45 + 25 + 25 + 45

= **180 km**

Displacement = PQ = **40 km**

Distance and displacement are **not equal**.

- Distance = PT =
**65 km**

Displacement = PT = **65 km**

Distance and displacement are **equal**.

- ‘A’ runs around the circular path of length 200 m five times. What are the distance and displacement covered by A.

The distance is **1000 m**, and the displacement is **0 m**, as A comes back to the same point.

**Speed and Velocity **

- The pace of motion of an object is called
**speed**of the object. However, the pace of motion along with its direction is called**velocity**.

- When a body covers equal distances in equal intervals of time, it is said to be in a
**uniform motion**. - When a body covers unequal distances in equal intervals of time, it is said to be in a
**non-uniform motion**. - The SI unit of measurement of speed and velocity is
**m/s**. However, it can also be measured in the units of**km/h, miles/h,**etc. - For a non-uniformly moving body, the
**average speed/velocity**is calculated.

**Mathematical Treatment: **

The **speed** of a body is mathematically given by,

**Speed = Distance/Time **

The** velocity **of a body is mathematically given by,

**Velocity = Displacement/Time **

The average **speed** of a body is mathematically given by,

**Average speed = Total distance/Total time taken **

The** average velocity **of a body moving with a uniformly increasing velocity is mathematically given by,

**Average velocity = (initial velocity + final velocity)/ 2 **

**Problems and Solutions **

- A car covers 1.8 km in a minute. What is the speed of the car?

**Solution: **

Distance covered = 1.8 km = 1.8 x 1000 = 1800 m

Time taken = 1 minute = 60 seconds

Therefore, speed = 1800/60

= **30 m/s**

**Thus, the speed of the car is 30 m/s. **

- An object travels first 20 m in 5 seconds and the next 20 m in 3 seconds. Calculate its average speed.

**Solution: **

Time taken to cover first 20 m = 5 seconds

Time taken to cover next 20 m = 3 seconds

Average Speed = Total distance covered/Total time taken

Average speed = (20 + 20) / (5 + 3)

= 40/8

= **5 m/s**

Thus, the average speed of the object is 5 m/s.

**Acceleration and Its Types **

- When a body undergoes a change in its velocity, it is said to be accelerating.
- When a body speeds up, it is said to be positively accelerating and when it slows down, it is said to be negatively accelerating.

- When a car’s acceleration remains constant throughout its motion, it is said to be uniformly accelerating. However, when its acceleration keeps varying throughout its motion, it is said to be non-uniformly accelerating.
- The SI unit of measurement of acceleration is m/s2.
- The
**direction**of acceleration is always towards the direction in which the velocity of a moving body increases. - Therefore, the
**positive acceleration**is always directed**towards**the direction of motion of the body. - However, the
**negative acceleration**is always directed**opposite**to the direction of motion of the body.

**Mathematical treatment: **

Let the initial velocity of a body at a certain instant of time be **u**. After a time, **t**, the final velocity of the body is found to be **v**. Then the acceleration of this body is mathematically given by,

**Acceleration = Change in velocity/Time taken **

** a = (v ****– ****u) / t **

**Problems and Solutions **

- The data of the varying velocities of a car is given in the table below. Calculate the change in velocity and identify the parts wherein the car has a positive acceleration and a negative acceleration.

**Solution: **

**Uniform Circular Motion: An Accelerated Motion **

When an object’s velocity increases or decreases by **equal amounts** in equal intervals of time, the acceleration of the object is **uniform**. Thus, an object is said to be in a **uniformly accelerated motion**.

When a body is in a uniform circular motion, its speed is the same throughout its motion. However, the direction of the body changes at every point in the motion. As velocity is a vector quantity that involves speed and direction of motion of the body, any change in either means a change in velocity. Therefore, the velocity changes at every point in a circular motion because of the change in direction. A change in velocity with time leads to an acceleration. Thus, a uniform circular motion is an accelerated motion. However, the acceleration here is directed towards the center of the circular path.

### Summary

1. Displacement is the shortest distance between two points.

2. Velocity is the rate of change of displacement.

3. Speed with direction is called velocity.

4. The direction of acceleration is always along the increasing velocity.

5. The SI unit of displacement, velocity and acceleration are m, m/s and m/s2 respectively.

6. Acceleration is positive when a body speeds up and is negative when the body slows down.

7. Positive acceleration is directed along the direction of velocity. In contrast, negative acceleration is directed opposite to the direction of the velocity.

8. A uniformly accelerated body changes its velocity by equal amounts in equal intervals of time.

9. A non-uniformly accelerated body changes its velocity by unequal amounts in equal intervals of time.

10. Uniform circular motion is an accelerated motion.

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