Motion

Motion is a continuous change in the position of the object with respect to the observer.

Types of Motion

1. Rectilinear Motion

When the position of moving objects is changing with respect to time along a straight line, it is called motion in a straight line or rectilinear motion. 

For example, a ball rolls on a horizontal surface, a stone falling from a building, a runner on 100 m race track, a vehicle moving on straight road.

2. Circular Motion

When an object follows a circular path during motion, it is called circular motion.

For example, motion of time hands of a clock, motion of child sitting on a merry-go-round, motion of the blades of an electric fan.

3. Oscillatory or Periodic Motion

When an object oscillates (to and fro) about a point (position of rest or equilibrium position), it is called oscillatory or periodic motion. An object repeats its motion after after a fixed interval of time.

For example, the motion of a child on swing, pendulum of wall clock, strings of a guitar.

Rolling Motion

The motion of a ball on the ground is rolling as it is rotating as well as moving forward on the ground. Thus, the ball undergoes a rectilinear motion as well as rotational motion.

Distance and Displacement

Motion is a change in the position of an object with time. The change in this position can be determined through distance measurements. This allows to know how fast or slow a motion is.

For a moving object two points are significant. One is the point of start or origin where from the object starts its motion and the other is the point where it reaches after certain interval of time. Points of start and destination are connected by a path taken by the object during its motion.

Distance

The length of the path followed by object is called distance. There may be a number of paths between the point of start and the point of destination. Hence the object may cover different distances between same point of start and destination. Distance is a scalar quantity.

Odometer measures the distance moved by the vehicle.

Displacement

In any motion, the object gets displaced while it changes its position continuously. The change in position of the object is called displacement. It is the shortest distance between initial and final position of the object. Displacement is a vector quantity. Its unit and dimensional formula are same as those of distance.

Time

Our ancestors noticed that many events in nature repeat themselves after definite intervals of time. For example, they found that the sun rises everyday in the morning. The time between one sunrise and the next was called a day. Similarly, a month was measured from one new moon to the next.

Often you need to measure intervals of time which are much shorter than a day. Clocks or watches are the most common time measuring devices. Periodic motion of a pendulum has been used to make clocks and watches.

Uniform and Non-Uniform Motion

The motion in which an object covers equal distance in equal interval of time is called uniform motion whereas the motion in which distance covered by object is not equal in equal interval of time is called non-uniform motion.

For the uniform motion, the graph is a straight line graph and for non-uniform motion, the graph is not a straight line.

Speed and Velocity

Speed

The distance moved by an object in a unit time is called its speed. The speed of an object is the distance traveled divided by the time taken to cover that distance. Its SI unit is metre per second (m/s). Other common unit is km/hr.

Distance covered = Speed × Time

If the speed of an object moving along a straight line keeps changing, its motion is said to be non-uniform. On the other hand, an object moving along a straight line with a constant speed is said to be in uniform motion.

Speedometer records the speed directly in km/h.

Velocity

Velocity is the speed of an object moving in a definite direction. The velocity of an object can be uniform or variable. It can be changed by changing the object’s speed, direction of motion or both.

When motion is along the shortest path, it is directed from the point of start to the point of finish. How fast this motion is determines the velocity. The velocity is the ratio of length of the shortest path (displacement) to the time taken.

Average Speed & Average Velocity

Speed during a certain interval of time cannot be used to determine total distance covered in given time of the journey and also the time taken to cover the total distance of journey. It is because a body does not always travel equal distance in equal interval of time. In most of the cases the body travels non-uniformly. Thus, in case of non-uniform motion to determine average speed is quite useful. The average speed can be determined by the ratio of total distance covered to the total time taken.

Average speed = Total distance traveled / Total time taken

v = s/t

Similarly, in case of average velocity in place of total distance covered you can take total displacement.

Relative Velocity

The relative velocity of an object B relative to a stationary or moving object A is equal to the time rate of change of position of object B with respect to object A.

If two objects A and B are moving with velocities u and v respectively in the same direction then the velocity of A relative to B will be (u - v) and the velocity of B relative to A will be (v - u).

If A and B are moving in opposite directions then the velocity of A relative to B will be (u + v).

Graphical Representation

To describe the motion of an object, you can use line graphs. It shows the change in one quantity corresponding to another quantity in the graphical representation.

Position-Time Graph / Distance-Time Graph

To draw graph of the motion of an object, its position at different times are shown on y-axis and time on x-axis.

For uniform motion position-time graph is a straight line. If the distance-time graph is a straight line, it indicates that the object is moving with a constant speed.

position time graph

Velocity-Time Graph

The variation in velocity with time for an object moving in a straight line can be represented by a velocity-time graph. Time is taken on the horizontal axis (x-axis) and velocity is on the vertical axis (y-axis).

If the object moves at uniform velocity, the height of its velocity-time graph will not change with time. It will be a straight line parallel to the x-axis.

for all uniformly accelerated motion, the velocity-time graph is a straight line. The area under the velocity-time graph gives the distance (magnitude of displacement) moved in a given interval of time.

The area enclosed by velocity-time graph and the time axis is equal to the magnitude of the displacement.

In the case of non-uniformly accelerated motion, velocity-time graphs can have any shape.

Acceleration

During uniform motion of an object along a straight line, the velocity remains constant with time. In this case, the change in velocity of the object for any time interval is zero.

The change in velocity with time is called acceleration. Thus, the acceleration of an object is defined as the change in velocity divided by the time interval during which this occurs.

This kind of motion is known as accelerated motion. The acceleration is taken to be positive if it is in the direction of velocity and negative when it is opposite to the direction of velocity.The unit of acceleration is ms-2.

Uniform Acceleration

If the acceleration of an object during its motion is constant, the object is moving with uniform acceleration. The motion of a freely falling body is an example of uniformly accelerated motion.The velocity-time graph of such a motion is straight line inclined to the time axis. 

However, in non-uniform acceleration, velocity varies with time. For example, if a car travelling along a straight road increases its speed by unequal amounts in equal intervals of time, then the car is said to be moving with non-uniform acceleration.

For a given time interval, if the final velocity is more than the initial velocity, then the acceleration will be positive. However, if the final velocity is less than the initial velocity, the acceleration will be negative.

Equations of Motion

Consider an object moving with uniform acceleration, a. Let u be the initial velocity (at time t = 0), v, velocity after time t and s, displacement during this time interval.

First Equation of Motion

Acceleration = Change in velocity / Time interval

a = (v - u)/t

v = u + at

Second Equation of Motion

Displacement = (average velocity) × (time interval)

s = (v + u)/2 × t

s = (u + at + u)/2 × t

s = ut + ½at2

Third Equation of Motion

a = (v - u)/t

s = (v + u)/2 × t

Multiply these two equations,

2as = v2 - u2

v2 = u2 + 2as

Uniform Circular Motion

When the velocity of an object changes, the object is accelerating. The change in the velocity could be due to change in its magnitude or the direction of the motion or both. When a particle moves with a constant speed in a circular path, its motion is called the uniform circular motion.

Circumference of a circle of radius r is given by 2πr.

v = 2πr/t

Angular Velocity

In rotational motion, the angle turned by a particle in a unit time is called its angular velocity. If in time t seconds the particle rotates through an angle θ radian, then 

Angular velocity, ω = θ/t (radian/sec)

If the particle completes one revolution in T seconds and it completes n revolutions in one second, then

ω = 2π/T

T is the time period and n is the frequency of rotation of particle.

Angular acceleration of the particle

α = ω/t

Linear Velocity

Linear velocity = Angular velocity x Radius

v = ω x r

Angular acceleration

α = a/r

where a is the linear acceleration of the particle.

Centripetal Acceleration

A particle moving with a uniform speed v in a circle experiences an acceleration v2/r directed towards the centre. This acceleration is called centripetal acceleration. The magnitude of centripetal acceleration always remains constant but its direction continuously changes. It is always directed towards the centre at every point of motion.

Centripetal Force

When a particle or a body moves with a uniform speed v on a circular path of radius r, then it is acted upon by a force mv2/r which is always directed towards the centre of the circle. This force is called centripetal force. The magnitude of this force remains constant but since this force is always directed towards the centre its direction continuously changes.

Simple Pendulum

A simple pendulum consists of a small metallic ball or a piece of stone suspended from a rigid stand by a thread. The metallic ball is called the bob of the pendulum. When the bob of the pendulum is released after taking it slightly to one side, it begins to move to and fro. The to and fro motion of a simple pendulum is an example of a periodic or an oscillatory motion.

The pendulum is said to have completed one oscillation when its bob, starting from its mean position O, moves to A, to B and back to O. The pendulum also completes one oscillation when its bob moves from one extreme position A to the other extreme position B and comes back to A. The time taken by the pendulum to complete one oscillation is called its time period.