**MOTION**

**INTRODUCTION**

Motion is a very preliminary state of action associated with living and non-living beings. The study of the displacement, velocity and acceleration associated with moving bodies can make us understand the motion of bodies. To have an in-depth study of motion, equational representation and graphical analysis of various related quantities in motion with time is also done.

When a body does not change its position with time, we can say that the body is at rest.

While if a body changes its position with time, it is said to be in motion.

(i) An object is said to be a point object if it changes its position by distances which are much greater than its size.

(ii) A point or some stationary object with respect to which a body continuously changes its position in the state of motion is known as origin or reference point.

**Describing Motion :**

When a tree, is observed by an observer A sitting on a bench, the tree is at rest. This is because position of the tree is not changing with respect to the observer A.

Now, When the same tree T is observed by an observer sitting in a superfast train moving with a velocity v, then the tree is moving with respect to the observer because the position of tree is changing with respect to the observer B.

**Vectors :** Physical quantities defined with both magnitude and direction are called vector quantities. They should also satisfy the law of vector addition.

**Examples:** Velocity, acceleration, force, displacement, momentum, weight, torque, electric field, magnetic filed, etc.

**Scalars :** Physical quantities having only magnitude are called scalar quantities.

**Examples:** Mass, time, distance, speed, work, power, energy, electric charge, area, volume, density, pressure, electric potential, temperature, etc.

**TYPES OF MOTION**

**(A) According to Directions**

**(i) One dimensional motion** is the motion of a particle moving along a straight line.

(ii) Two dimensional motion A particle moving along a curved path in a plane has 2-dimensional motion.

(iii) Three dimensional motion Particle moving in space has 3-dimensional motion.

**(B) According to state of motion**

A moving body may cover equal distances in equal intervals of time or different distances in equal intervals of time. On the basis of above assumption, the motion of a body can be classified as uniform motion and non-uniform motion.

**Example of non-uniform motion –**

(i) An aeroplane running on a runway before taking off.

(ii) A freely falling stone under the action of gravity.

(iii) An object thrown vertically upward.

(iv)When the brakes are applied to a moving car.

**(C) According to path**

**(i) Linear motion :** A body has linear motion if it moves in a straight line or path.

**Ex.**(a) Motion of a moving car on a straight road.

(b) Motion of a ball dropped from the roof of a building.

**Motion Summary**

**Motion (Uniform Motion And Non Uniform Motion, Acceleration and Velocity)**

•A particle is a point-like object, has mass but infinitesimal size

• The object’s position is its location with respect to a chosen reference point, In the diagram, the road sign the reference point

• Motion occurs when an object changes its position.

Both Distance and Time are important in describing motion.

• Sometimes you know motion has occurred even if you didn’t see it happen. (mail truck)

• Relative motion: when two objects are moving in a plane (either in same direction or opposite) each have relative motion with respect to second. e.g. a person sitting in a train and watching a tree, in this case tree is stable but is assumed to be moving but with respect to train.

**Distance vs. Displacement**

• **Distance:** How far an object has moved. It has only magnitude without direction. (total)

• **Displacement:** How far and in what direction an object has moved from its start position. i.e. the direct distance between two points.

**Speed**

•Speed = the distance an object travels in a given amount of time

•Speed = distance/time

•sI unit of speed is m/s

**Types of Speed**

•Constant speed: speed doesn’t change (set your car on cruise control)

•Changing speed: Riding a bike for 5 km. Take off and increase speed, slow down up hill, speed up down hill, stop for stop sign. The trip took you 15 min (.25 h)

•Average speed:

•Instantaneous speed: speed at any given time.

**Velocity**

•Velocity: includes speed and DIRECTION

•Storm is moving at 20km/hr.

•Should you be seeking shelter?

•Suppose two trains are going with the same speed in opposite direction so they are having different velocities.

•Race car going around an oval track might have constant speed, but different velocities at each point.

**Acceleration**

•Any change in velocity over a period of time is called acceleration.

•The sign (+ or -) of indicates its direction. + sign shows the acceleration and – sign shows de-acceleration.

•Uniform (constant) acceleration equation

• Images of car are equally spaced.

• The car is moving with constant positive velocity (shown by red arrows maintaining the same size) .

• The acceleration equals to zero

• Images of car become farther apart as time increases

• Velocity and acceleration are in the same direction

• Acceleration is uniform (Arrows below the car maintain the same length)

• Velocity is increasing (Arrows above the car are getting longer)

• This shows positive acceleration and positive velocity

The instant speed at points of equal elevations is the same.

The velocities are different because they are in opposite Free Fall & Air Resistance

**Galileo Galilei Italian physicist and astronomer**

**Formulated laws of motion for objects in free fall**

• A freely falling object is any object moving freely under the influence of gravity alone.

• It does not depend upon the initial motion of the object

• Dropped – released from rest

• Thrown downward

• Thrown upward

• The acceleration of an object in free fall is directed downward, regardless of the initial motion

• The magnitude of free fall acceleration (gravitational acceleration) is g = 9.80 m/s2

• g decreases with increasing altitude

• g varies with latitude, height and depth from earth surface.

• 9.80 m/s2 is the average at the Earth’s surface

• The italicized g will be used for the acceleration due to gravity

• Not to be confused with g for grams

• With negligible air resistance, falling objects can be considered freely falling.

objects of different shapes accelerate differently (stone vs feather)

• Speed both upward and downward

• The path is symmetrical.

• Acceleration is constant.

• The magnitude of the velocities is the same at equal heights.

• Images become closer together as time increases

• Acceleration and velocity are in opposite directions when ball goes upward.

• Acceleration is uniform (violet arrows maintain the same length)

• Velocity is decreasing in upward motion (red arrows are getting shorter)

• Positive velocity and negative acceleration

• Velocity becomes zero at maximum height.

• Time duration flight in going upward and coming back is always same.

**Test Yourself :**

1. What is SI Unit of displacement?

2. Name the quantity which represents rate of change of velocity.

3. A particle describes a semicircle of radius l 14m. What are its distance and displacement covered?

**2Graphical Representation Of Motion & Graphs**

**Test Yourself :**

1. What does slope of Position – Time graph represent?

2. If velocity –time graph is parallel to time axis, what type of motion does it represent?

3 Equation of motion

(1) When object is moving in straight line-

• v = v_{o}+ at

• x = x_{o}+ v_{o}t + ½ at^{2}

• v^{2}= v_{o}^{2}+ 2a(Δx)

• Average acceleration describes how fast the velocity is changing with respect to time.

• where:

a_{ave}= average acceleration

• v = change in velocity

• x = displacement

• t = elapsed time

(2) when object is coming vertically downward-

• v = v_{o}+ gt

• h = v_{o}t + ½ gt^{2}

• v^{2}= v_{o}^{2}+ 2ah

(3) when object is coming vertically upwardv

= v_{o}- gt

h = v_{o}t - ½ gt^{2}

v_{2}= v_{o}^{2}- 2gh

• The SI unit of velocity is the m/s.

Average accleration is + or – depending on direction.

• Instantaneous Acceleration

• Instantaneous acceleration is the limit of Δv/Δt as Δt approaches zero.

• Instantaneous acceleration is zero where slope is constant

• Instantaneous acceleration is positive where curve is concave up

• Instantaneous acceleration is negative where curve is concave down

**Test Yourself :**

1. Give the equation for uniform motion.

2. When a car stops after applying brakes, what is the final velocity?

**4 Uniform Circular Motion**

In this kind of motion the object moves on circle with fix speed but the direction is changed by the time so the velocity of the change so its called acceleration motion. This acceleration is called centrifugal acceleration. It is directed toward the centre.

**Test Yourself:**

1. What remains constant in uniform circular motion?

2. What changes continuously in uniform circular motion?

**Key Learnings:**

**1**. If the position of an object does not change with time, it is said to be at rest.

**2**. If the position of an object changes as time passes, it is said to be in motion.

**3**. Reference point is a fixed point with respect of which a body is at rest or in motion.

**4**. Rest and Motion are relative terms.

**5**. Distance is the length of actual path traveled by a body in a given time.

**6**. Displacement is the shortest distance between the initial and final positions of the body in a known direction.

**7**. A physical quantity which has both magnitude and direction is called as vector quantity.

**8**. A physical quantity which has only magnitude is called as scalar quantity.

**9.**The S.I unit of distance and displacement is metre.

**10**. A body is said to be in uniform motion, if travels equal distances in equal intervals of time.

**11**. A body is said to have non-uniform motion if it travels unequal distances in equal intervals of time.

**12**. Speed is the ratio of distance traveled to the time taken to coverthat distance.

**13**. In non-uniform motion, speed of an object is not constant. The S.I. unitof speed is m/s or ms-1.

**14**. Average speed of a body is the total distance traveled divided by the total time taken.

**15**. Velocity is displacement per unit time. The S.I. unit of velocity is meter per second.

**16**. Average velocity is displacement divided by the time taken.

**17**. Speed is a scalar quantity and velocity is a vector quantity.

**18**. Time is independent variable, plotted along X-axis. Distance is dependent variable, plotted along Y-axis.

**19**. Graphs are designed to make it easier for the reader to interpret and understand numerical data.

**20**. The distance-time graph is a straight line parallel to time axis when the object is at rest.slope a straight line= (y_{2}-y_{1)}/ (x_{2}-x_{1})

**21**. The nature of distance-time graph is a straight line when the object is inthe state of uniform motion.

**22**. Slope of the distance-time graph gives the speed of the obje

**23**. A more steeply inclined distance-time graph indicates greater speed. The nature of distance-time graph is a curve having varying slope when the object has non-uniform moti

**24**. If the velocity of a body remains constant, the velocity-time graph is a horizontal line parallel to the time axis

**25**. If the velocity of the body changes uniformly at a constant rate, the velocity-time graph is a straight line.

**26**. If the velocity of the object changes non-uniformly, the velocity-time graph is a curve having increasing slope.

**27**. The area enclosed by the velocity-time graph and the time axis represents the displacement.

**28**. The slope of the velocity-time graph gives the acceleration

**29**. When a body travels along a circular path of constant radius with a constant speed v then its motion is uniform circular motion

**30**. In a uniform circular motion, velocity of a particle is not constant but its speed is constant, hence it is an accelerated motion

**T****op Formulae**

**1****.**Average Speed = Total dis tan ce travelled / Total time taken

If an object travels a distance s in time t then its speed v is given by

v = s/t

**2****.**Average Velocity (V_{av}) = ( initial velocity (u)+ final velocity (v) ) / 2

**3****.**Acceleration = change in velocity / time taken

If the velocity of an object changes from an initial value u to the final value v in time t, the acceleration a = (v -u)/t

**4****.**Three equations of motion

v =u +at.........(1)

s =ut +1/2 at^{2}.......(2)

v^{2}=u^{2}+2aS........(3)

Where u is the initial velocity of the object which moves with uniform acceleration a for time t. v is the final velocity, and s is the distance traveled by the object in time t.

**5****.**We know that the circumference of a circle of radium r is given by 2πr. If a person takes t seconds to go once around the circular path of radius r, the velocity v is given by v =2πr/t

**Q1 When is the body said to be in motion?**

Ans.A body is said to be in motion if its position change sw.r.t.a reference point.**Q.2 Motionis a relative term. Explain?**

Ans.Motion is a relative term becausea body can be i nmotion w.r.t.one set ofsurroundings and at the same time can be at rest w.r.t. another set of surroundings.eg. a person sitting in a moving car is at rest w.r.t his co-passengers where as he isin motion w.r.t. person standing outside.

**Q.3 Differentiate between scalar and vector quantities.**

Ans.Scaler quantities are those which can be described by their magnitude only.e.g.Mass,time,temperature,length,distance,speed etc.Vector quantities are those quantities which need both magnitude anddirection for their description.e.g.displacement velocity,acceleration,force etc.

**Q.4 Differentiate between distance and displacement?**

Ans.Distance is the length of th eactual path covered by the body.it is a scalarquantity.Dispalacement is the shortest dis tance from the initial to the final position.itis thevector quantity.

**Note**that the magnitude of the displacement for a course of motion may be zero but the corresponding distance covered is not zero.

**Q5. What is the distance covered and the displacement ofa body moving along asemi-circular path of a radius"R".**

**Ans.**Distance=22/7R

Displacement=2R

**Q.6 Differentiate between speed and velocity.**

Ans. Speed is the distance covered by a bodyper unit time. It is a scaler quantity.Velocity is the speed of a body in a particular direction. It is a vectorquantity.

**Q7. Differentiate between Uniform and Non-uniform Motion?**

Ans.**Uniform Motion**-if body cover sequal distance in equa linterva lsoft imein a straight line, it is said to be in uniform motion.e.g light has a uniform motion.

**Non Uniform motion**-if a body covers unequal distance in equal intervalsof time,it is said to haven on-uniform motion.e.g.a freely falling body,a carmoving in a crowded road.

**Q.8 Define Average Speed?**

ans. For a bod yhaving non-uniform motion,

average speed=total distance travelled/ total time taken.

**Q9.****Define Acceleration?**

Ans.Acceleration is defined as rate of change of velocity.

a=v-u/t where a stands for acceleration

v-final velocity

U-initial velocity

t-time

The SI unit of acceleration is m/s^{2}

**Q10 What is meant by uniform and non uniform acceleration?**

Ans If an object travels in a straight line and its velocity increases or decreases by equal amounts in equal intervals of time, then the acceleration of the object is said to be uniform. The motion of a freely falling body is an example of uniformly accelerated.

On the other hand, an object can travel with non-uniform acceleration if its velocity changes at a non-uniform rate. 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.

**NOTE**If a body moves with a velocity of constant magnitude along the circular path, the only change in its velocity is due to the change in the direction of motion. The motion of the body moving along a circular path is, therefore, an example of an accelerated motion.

**Q11 Draw distance-time grashowing unifonn and non uniform**

**Q12 Draw speed-time grashowing unifonn and non unifonn motion.**

**NOTE**The area enclosed by velocity-time graph and the time axis will be equal to the magnitude of the displacement.

**Q13 Derive the three equations of motion.**

**FIRST EQUATION OF MOTION**

From this graph, you can see that initial velocity of the object is u (at point A) and then it increases to v (at point B) in time t. The velocity changes at a uniform rate a. interval t.

Let us draw AD parallel to OC. From the graph, we observe that

BC = BD + DC = BD + OA

Substituting BC = v and OA = u,

we get v = BD + u

or BD = v . u

From the velocity-time graph the acceleration of the object is given by a =Change in velocity/ time taken

=BD/AD

= BD/OC

Substituting OC = t, we get a =BD/ t

or BD = at (8.9)

v = u + at

**SECOND EQUATION OF MOTION**

the distance s travelled by the object is given by

s = area OABC (which is a trapezium)

= area of the rectangle OADC + area of the triangle ABD

**THIRDEQUATION OF MOTION**