Introduction
Motion is found everywhere in nature. Examples include butterflies flying, snakes slithering, horses galloping, dust particles moving in sunlight, clouds gathering, tides rising and falling, and planets moving in space.
Since many motions around us are complex, scientists first study motion in simpler forms. In this chapter, we mainly study:
- Motion in a straight line
- Graphical representation of motion
- Kinematic equations
- Motion in a plane
- Uniform circular motion
Motion can be described using words, numbers, equations, and graphs.
4.1 Motion in a Straight Line
When an object moves along a straight path, its motion is called linear motion or motion in a straight line.
Examples
A car moving on a straight road, a train moving on a straight track, a ball falling vertically, and swimmers moving in a straight lane are examples of motion in a straight line.
4.1.1 Describing Position
To describe the position of an object, we need a reference point.
A reference point is a fixed point with respect to which the position of an object is described.
Position
The position of an object is described by:
- Its distance from the reference point
- Its direction from the reference point
For example, if a student is 20 m to the right of a tree, then the tree can be taken as the reference point.
Usually, positions to the right of the reference point are taken as positive, and positions to the left are taken as negative.
Rest and Motion
An object is said to be in motion if its position changes with time with respect to a reference point.
An object is said to be at rest if its position does not change with time with respect to a reference point.
Important Point
Rest and motion depend on the reference point. For example, a person sitting inside a moving bus is at rest with respect to another passenger, but in motion with respect to a person standing on the road.
4.1.2 Distance Travelled and Displacement
Distance Travelled
The distance travelled by an object is the total length of the path covered by it.
Features of Distance
Distance has only magnitude. It does not need direction. Therefore, it is a scalar quantity.
SI unit of distance = metre (m)
Example: If a person walks 100 m forward and then 60 m backward, total distance travelled = 100 m + 60 m = 160 m.
Displacement
Displacement is the net change in the position of an object between two instants of time.
It is the shortest distance between the initial and final positions, along with direction.
Features of Displacement
Displacement has both magnitude and direction. Therefore, it is a vector quantity.
SI unit of displacement = metre (m)
Example: If an athlete starts from point O, goes to point A at 100 m, and comes back to point B at 40 m, then:
Distance travelled = 100 m + 60 m = 160 m
Displacement = 40 m in the positive direction
Difference between Distance and Displacement
| Basis | Distance | Displacement |
|---|---|---|
| Meaning | Total path covered | Net change in position |
| Direction | No direction | Has direction |
| Type | Scalar | Vector |
| Value | Always positive or zero | Can be positive, negative, or zero |
| Path dependence | Depends on path | Depends only on initial and final positions |
| Unit | metre | metre |
Important Points
The magnitude of displacement is always less than or equal to the total distance travelled.
Displacement can be zero even when distance travelled is not zero. For example, when a person starts from home, goes to a shop, and returns home, displacement is zero but distance is not zero.
For motion in a straight line without turning back, distance travelled and magnitude of displacement are equal.
4.1.3 Average Speed and Average Velocity
Average Speed
Average speed tells us how fast or slow an object moves.
Average speed=Time intervalTotal distance travelled
Average speed has only magnitude, so it is a scalar quantity.
SI unit = m s⁻¹ or m/s
It can also be measured in km h⁻¹.
Uniform Motion in a Straight Line
If an object travels equal distances in equal intervals of time, it is said to be in uniform motion.
Example: A car covers 20 m in every second.
Non-uniform Motion in a Straight Line
If an object travels unequal distances in equal intervals of time, it is said to be in non-uniform motion.
Example: A car covers 5 m in the first second, 10 m in the second second, and 20 m in the third second.
Average Velocity
Average velocity describes how fast the position of an object changes and in which direction.
vav=ts
Here,
vav = average velocity
s = displacement
t = time interval
Average velocity has both magnitude and direction, so it is a vector quantity.
SI unit = m s⁻¹
Difference between Average Speed and Average Velocity
| Basis | Average Speed | Average Velocity |
|---|---|---|
| Formula | Total distance ÷ time | Displacement ÷ time |
| Direction | No direction | Has direction |
| Type | Scalar | Vector |
| Can be zero? | Only if distance is zero | Can be zero even when distance is not zero |
| Unit | m/s | m/s |
Example
If Sarang swims from one end of a 25 m pool to the other end and comes back in 50 s:
Total distance = 50 m
Displacement = 0 m
Average speed = 50 ÷ 50 = 1 m/s
Average velocity = 0 ÷ 50 = 0 m/s
4.1.4 Average Acceleration
Velocity of an object may change with time. This change in velocity is described by acceleration.
Average Acceleration
Average acceleration is the change in velocity divided by the time interval.
a=tv−u
Here,
a = average acceleration
u = initial velocity
v = final velocity
t = time interval
SI unit of acceleration = m s⁻² or m/s²
Direction of Acceleration
If speed increases, acceleration is in the direction of velocity.
If speed decreases, acceleration is opposite to the direction of velocity. This is also called negative acceleration or retardation.
Important Point
An object can move very fast and still have zero acceleration if its velocity is constant.
Example: A bus moving at constant velocity on a straight road has zero acceleration.
Acceleration Due to Gravity
When an object is dropped from a height, its velocity increases every second due to Earth’s gravitational force.
The acceleration due to gravity is denoted by g.
Near the surface of Earth:
g = 9.8 m s⁻²
4.2 Graphical Representation of Motion
Graphs are useful for representing motion visually. They help us understand how position, velocity, and acceleration change with time.
In this chapter, graphs are mainly used for motion in a straight line in one direction.
4.2.1 Plotting Graph
To plot a graph:
- Draw two perpendicular lines.
- The horizontal line is the x-axis.
- The vertical line is the y-axis.
- Their point of intersection is called the origin.
- Choose a suitable scale.
- Plot the points using the given data.
- Join the points to get the graph.
For motion graphs:
Usually, time is taken on the x-axis.
Position, velocity, or acceleration is taken on the y-axis.
Important Note
A graph is not a route map. It does not show the actual path of the object. It shows how one quantity changes with another quantity.
4.2.2 Position-Time Graphs
A position-time graph shows how the position of an object changes with time.
Time is taken on the x-axis.
Position is taken on the y-axis.
Shape of Position-Time Graph
1. Straight Line Position-Time Graph
A straight line position-time graph shows that the object is moving with constant velocity.
The object covers equal displacements in equal time intervals.
2. Curved Position-Time Graph
A curved position-time graph shows that the velocity is changing.
This means the object is in accelerated motion.
3. Horizontal Position-Time Graph
A straight line parallel to the time axis shows that the object is at rest.
Its position is not changing with time.
Velocity from Position-Time Graph
The slope of a position-time graph gives velocity.
Velocity=Change in timeChange in position
A steeper slope means greater velocity.
A smaller slope means lower velocity.
A horizontal line has zero slope, so velocity is zero.
4.2.3 Velocity-Time Graphs
A velocity-time graph shows how the velocity of an object changes with time.
Time is taken on the x-axis.
Velocity is taken on the y-axis.
What does the shape of the velocity-time graph indicate about the nature of motion?
1. Horizontal Velocity-Time Graph
If the velocity-time graph is a straight line parallel to the time axis, velocity is constant.
Acceleration = 0
This represents uniform motion.
2. Straight Line Sloping Upward
If the velocity-time graph is a straight line sloping upward, velocity increases by equal amounts in equal intervals of time.
Acceleration is constant and in the direction of velocity.
3. Straight Line Sloping Downward
If the velocity-time graph is a straight line sloping downward, velocity decreases by equal amounts in equal intervals of time.
Acceleration is constant and opposite to the direction of velocity.
Which physical quantities can be obtained from a velocity-time graph?
Two important physical quantities can be obtained from a velocity-time graph:
1. Acceleration
The slope of a velocity-time graph gives acceleration.
Acceleration=Change in timeChange in velocity
If the slope is positive, acceleration is positive.
If the slope is negative, acceleration is negative.
If the slope is zero, acceleration is zero.
2. Displacement
The area under a velocity-time graph gives displacement.
For constant velocity:
Displacement = velocity × time
For uniformly accelerated motion, the area may be found by dividing the graph into simple shapes like rectangles and triangles.
4.3 Kinematic Equations for Motion in a Straight Line with Constant Acceleration
Kinematic equations are mathematical equations used to describe motion in a straight line with constant acceleration.
These equations relate five physical quantities:
- s = displacement
- t = time interval
- u = initial velocity
- v = final velocity
- a = acceleration
First Kinematic Equation
v=u+at
This equation is used to find final velocity when initial velocity, acceleration, and time are known.
Second Kinematic Equation
s=ut+21at2
This equation is used to find displacement when initial velocity, acceleration, and time are known.
Third Kinematic Equation
v2=u2+2as
This equation is used when time is not given or not required.
Important Points about Kinematic Equations
These equations are valid only when acceleration is constant.
In motion in one direction, distance and magnitude of displacement are equal.
In motion involving both directions, signs of u, v, a, and s show direction.
Example
A car moving at 54 km/h applies brakes and comes to rest. If acceleration is –4 m/s², find the stopping distance.
Convert speed:
54 km/h = 15 m/s
Given:
u = 15 m/s
v = 0 m/s
a = –4 m/s²
s = ?
Using:
v² = u² + 2as
0² = 15² + 2 × –4 × s
0 = 225 – 8s
8s = 225
s = 28.125 m
So, the car travels about 28.1 m before stopping.
4.4 Motion in a Plane
Motion in a plane is motion in two dimensions.
Examples:
A ball kicked in air, a vehicle overtaking another vehicle, and a satellite moving around Earth are examples of motion in a plane.
Motion in a straight line is one-dimensional motion.
Motion in a plane is two-dimensional motion.
4.4.1 Uniform Circular Motion
When an object moves in a circular path, its motion is called circular motion.
When an object moves in a circular path with constant speed, its motion is called uniform circular motion.
Examples
A child sitting on a merry-go-round, a stone tied to a string and whirled in a circle, the tip of a clock hand, and a planet revolving around the Sun are examples of circular motion.
Distance and Displacement in Circular Motion
If an object completes one full revolution in a circular path:
Distance travelled = circumference of the circle = 2πR
Displacement = 0
This is because the object returns to its starting point.
Speed in Uniform Circular Motion
vav=T2πR
Here,
R = radius of circular path
T = time taken for one revolution
Velocity in Uniform Circular Motion
In uniform circular motion, speed remains constant, but velocity changes continuously.
This happens because velocity has direction, and the direction of motion changes at every point on the circle.
At any point on a circular path, velocity is along the tangent to the circle at that point.
Acceleration in Uniform Circular Motion
Uniform circular motion is accelerated motion because the direction of velocity changes continuously.
Even if speed is constant, acceleration is present due to change in direction of velocity.
Important Point
In daily life, we often think acceleration means only increase or decrease in speed. But acceleration can also occur due to change in direction.
Important Formulae
| Quantity | Formula |
|---|---|
| Average speed | Total distance travelled ÷ Time interval |
| Average velocity | Displacement ÷ Time interval |
| Average acceleration | Change in velocity ÷ Time interval |
| First kinematic equation | v = u + at |
| Second kinematic equation | s = ut + ½at² |
| Third kinematic equation | v² = u² + 2as |
| Distance in one circular revolution | 2πR |
| Speed in uniform circular motion | 2πR ÷ T |
Important Terms
Motion
An object is in motion if its position changes with time with respect to a reference point.
Rest
An object is at rest if its position does not change with time with respect to a reference point.
Reference Point
A fixed point used to describe the position of an object.
Distance
The total path covered by an object.
Displacement
The net change in position of an object.
Speed
Distance travelled per unit time.
Velocity
Displacement per unit time.
Acceleration
Rate of change of velocity.
Uniform Motion
Motion in which an object covers equal distances in equal time intervals.
Non-uniform Motion
Motion in which an object covers unequal distances in equal time intervals.
Uniform Circular Motion
Motion of an object in a circular path with constant speed.
At a Glance — Summary
Motion is the change in position of an object with time with respect to a reference point. To describe the position of an object, we need both distance and direction from the reference point.
Distance is the total path covered by an object, while displacement is the net change in its position. Distance is a scalar quantity, whereas displacement is a vector quantity.
Average speed is calculated by dividing total distance travelled by time interval. Average velocity is calculated by dividing displacement by time interval. Average acceleration is the change in velocity divided by time interval.
Graphs are useful tools for representing motion. A position-time graph shows how position changes with time. Its slope gives velocity. A velocity-time graph shows how velocity changes with time. Its slope gives acceleration, and the area under it gives displacement.
For motion in a straight line with constant acceleration, the three kinematic equations are:
v = u + at
s = ut + ½at²
v² = u² + 2as
These equations are used to calculate displacement, velocity, acceleration, or time when the motion has constant acceleration.
Motion in a plane is motion in two dimensions. Uniform circular motion is a special type of motion in which an object moves in a circular path with constant speed. In this motion, speed remains constant, but velocity keeps changing because direction changes continuously. Therefore, uniform circular motion is accelerated motion.
Thus, this chapter helps us understand motion using basic quantities such as distance, displacement, speed, velocity, acceleration, graphs, and equations.
