let's take a look at some familiar [Music] objects when i move among these objects there is no change in the position of these objects that's because these objects are at rest now let's see what happens when these objects move wow did you see how fast the car zoomed away [Music] the ball has also disappeared out of sight the globe shows how the earth rotates on its axes but the rocking chair is still here its movement looks very inviting i'd like to relax in this chair so what changed around me the objects that were at rest started moving or were in motion motion can be of various types it can be regular and controlled like in a watch or erratic and uncontrolled like that displayed by the tsunami that took thousands of lives now how far do you think the car will go how long will the globe take to rotate completely on its axis you can obtain answers to such questions through the study of motion in physics these answers will help you control motion and harness it for constructive purposes like in a hydroelectric dam in this lesson you will learn about motion and related quantities at the end of this lesson you will be able to define motion identify instances of motion encountered in our everyday life analyze motion along a straight line identify uniform and non-uniform motion calculate speed and average speed define velocity calculate velocity of a moving object explain scada and vector quantities define acceleration and calculate acceleration of a moving object in everyday life you see some objects at rest and others in motion birds fly horses gallop water flows through rivers and cars move you say these are in motion thus motion is a state of an object in which there is a change in its position with respect to its surroundings and time when a car race starts the position of each car participating in the race changes relative to the stands spectators and so on this is because the cars ever evidence state of motion tiny particles such as atoms and molecules and heavenly bodies such as planets stars and galaxies are all in motion for example earth rotates around its own axis and revolves around the sun in an orbit similarly other planets solar systems and galaxies in the universe are all constantly in motion none of the moving objects that we have looked at exist in isolation they may have other moving or stationary objects in their environment what happens when we compare the state of two or more than two objects for instance let us consider the state of a car driver is he at rest or in motion it is difficult to answer this question this is because with respect to the car the driver is at rest but with respect to the other things around he is in motion thus the states of rest and motion of bodies are relative absolute states of rest or motion do not exist consider a bus moving along a road the passenger in the bus perceives the trees and other objects outside the bus to be moving while his co-passengers seem to be at rest similarly we cannot see the rotation or evolution of the earth however based on the changes in the environment at various times of the day and transition of seasons we can infer that the earth is rotating on its axis as well as revolving around the sun hence overall you are in motion as a part of earth but at the same time you are at rest with respect to your immediate surroundings motion of objects or bodies can be one-dimensional or linear two-dimensional or three-dimensional one-dimensional or linear motion involves an object moving along a straight line or in a specified direction for example cars moving along a straight path in the circuit exhibit linear motion in two-dimensional motion in a given plane an object moves in two directions at the same time for example a golf ball when hit flies off making a large parabolic part over the gulf coast it travels in two dimensions or directions at the same time horizontal and vertical if we observe an airplane taking off from a runway it ascends rapidly in one direction and after gaining some height turns towards its flight path this type of motion where an object moves in three directions at the same time is called three-dimensional motion in this lesson we will cover motion along a linear path in detail you can measure the magnitude of movement of an object using two quantities distance and displacement hi i am called displacement i will always show you the shortest distance between the initial and final points or the aerial distance between two points my cousin is called distance we often look similar so it can be a little confusing for people at times however there is a unique feature that sets me apart from my cousin my sense of direction in fact because of this characteristic i always display a special bowl letter or an arrow on my head let's look at these quantities using an example at the end of her school day mary goes from the school to the playground the school is three kilometers away from her home and the playground is four kilometers away from the school so from home to school to the playground mary travels a total of seven kilometers this is the distance traveled by mary thus distance is the actual length of the path covered by a moving object irrespective of the direction of motion however the straight route from mary's home to the playground is only about five kilometers long therefore mary's displacement from home to the playground is five kilometers as against the distance covered by mary in reaching the playground which was seven kilometers thus displacement is the shortest distance between two points irrespective of the path between the points when an object is in motion its motion may or may not be consistent all through its journey until it comes to rest again let us observe the yellow car in the race what do we see the car moves 50 meters every second continuously thus for each second if we know down how far the car has moved from the starting line we see that the displacement during any two consecutive seconds is the same this car is in uniform motion an object is said to be in uniform motion if it shows equal displacements in equal intervals of time however small these intervals may be now let us observe the red car in the rays and record how far it is moved from starting line for each second of motion we see that its displacement during any two consecutive seconds is not the same this is non-uniform motion an object is said to be a non-uniform motion if it travels unequal distances in equal intervals of time or equal distances in unequal intervals of time different objects may take different amounts of time to cover a given distance how fast does a moving object change its position the answer to this question relates motion to the time taken for the change of position and that brings us to speed speed of an object is the rate at which an object covers a given distance or changes its position speed is calculated as a ratio of distance covered to the time taken to cover that distance thus an object that covers a relatively large distance in a short amount of time is moving at a high speed conversely a slow-moving object has a low speed and covers a relatively small amount of distance in a relatively longer duration an object at rest has no speed speed is measured in centimeters per second in the cgs system and as meters per second in the si system typically you measure speed in kilometers per hour one kilometer per hour equals 5 divided by 18 meters per second or 1 meter per second equals 3.6 kilometers per hour let's try applying this formula to one of the situations we reviewed earlier in our previous example mary's school is three kilometers away from her home and the playground is four kilometers away from the school if mary walks to school from home in 20 minutes her speed can be calculated as 3 000 divided by 1200 which is 2.5 meters per second or 9 kilometers per hour however generally the motion of an object is not uniform from its starting point to its final destination its speed varies at different points or stretches in the journey in such cases to determine the speed at which the object covers the total distance you may need to gather data on the speed of the object at each instant when it is in motion and then calculate the average of all these instantaneous speeds however gathering so much of data can be tedious and certainly impossible in the case of large distances therefore we typically use a quantity called average speed in our calculations the average speed of an object is the ratio of the total distance covered by the object to the total time taken by it for covering that distance like motion speed can be uniform or non-uniform thus an object in uniform motion has uniform speed and an object in non-uniform motion has known uniform speed if you rearrange the formula for speed you can determine the distance covered in a given period of time at a given speed s is equal to d divided by t therefore d is equal to s multiplied by t however it does not give you the displacement of the object to determine the displacement of an object you need to know the direction of motion along with the rate of the motion velocity is the speed of an object moving in a definite direction velocity of an object is defined as the rate of change in the position of a body can also be defined as the rate of displacement of an object velocity is calculated as a ratio of displacement of the object to the time interval taken for that displacement velocity is measured as centimeter per second in the cgs system and meter per second in the si system similar to speed velocity is also measured in kilometers per hour velocity of an object can change by either changing its speed in a particular direction or by keeping the speed constant and changing the direction when the yellow car takes a turn and moves ahead its velocity becomes variable like average speed you can calculate the average velocity of an object having non-uniform motion average velocity is defined as the ratio of the total displacement to the total time taken when a body is moving with variable velocity for t total time duration u and v are the magnitudes of its initial and final velocities during which it has a total displacement that is s total the average velocity of the body is the average of u and v that is it is half the sum of initial and final velocities hi can you make out the change in the velocity of my hands as they work round the clock to tell you the time the tip of each hand moves with uniform speed but changes direction continuously as it moves in a circular path thus if you consider the second's hand its velocity actually changes each second when the seconds hand completes one full rotation in one minute it comes back to its original position at this point its displacement becomes zero and hence the average velocity also becomes zero so far we have reviewed some related measures of motion that is distance and displacement and speed and velocity among these measures distance and speed have only magnitude whereas displacement and velocity have magnitude as well as direction a physical quantity that has only magnitude but no direction is scalar length area distance and speed are all scalar quantities on the other hand a physical quantity that has both magnitude and direction is vector a vector such as displacement is simply a scalar with direction let's look at a car in the rays a yellow car is currently at a distance of seven kilometers from the starting line however if we measure the straight line from the starting line to the position of the yellow car the car is positioned at a point just five kilometers towards the northeast direction here distance of seven kilometers a scalar whereas the displacement of five kilometers towards the northeast is a vector for an object in non-uniform motion its velocity may change with time rate of change of velocity is expressed in terms of acceleration thus if a body moves in such a way that its velocity changes then it is said to be accelerating observe the car moving swiftly in the circuit when it approaches the turn it slows down and after negotiating the turn quickly gains velocity again this change in velocity of the car is known as its acceleration acceleration is the ratio of the difference of final and initial velocity to the time interval during which the velocity has changed [Music] acceleration is a vector quantity and is measured as centimeter per second square in the cgs system and as meter per second square in the si system like speed and velocity the acceleration of an object can be uniform or non-uniform a freely falling body gains velocity due to gravitational forces at a uniform rate thus it shows uniform acceleration however a car negotiating a turn shows non-uniform acceleration because its velocity changes at a non-uniform rate when an object moves such that its final velocity is less than its initial velocity it is said to possess negative acceleration or retardation here the two bassets a and b are moving along a straight path the bus a is gaining velocity hence it is accelerating whereas the bus b is losing velocity and hence it is retarding for example consider a bus traveling with the velocity of 20 meters per second the bus comes to a halt at a bus stop over 5 seconds in this case the acceleration of the bus is minus four meters per second square thus we say the retardation of the bus is equal to four meters per second square summarize let's take another look at the relationships between the quantities we just studied distance represents the extent or length of motion when distance is mentioned with the direction you get displacement a change in distance with respect to time gives you speed of motion similarly the rate of change of displacement gives you velocity when you measure the change in velocity with respect to time you get acceleration the section on solved problems provides you an opportunity to review some model problems based on these concepts to revisit the key points covered in this lesson please review the flash card remember the hair and tortoise story your granny read to you in your childhood the story goes like this o'hare was watching the slow progress of a tortoise on a path in the jungle being a faster runner himself he began to make fun of the tortoise the tortoise challenged the hair to erase and the hair accepted confidently when the race began the hare shot ahead and ran briskly for some time then seeing that he was far ahead of the tortoise he decided to sit and relax for a while before continuing the race he soon fell asleep the tortoise plodding on overtook him and finally won the race the table provided shows the data on their progress during the race similarly the graph represents the same data can you see how the hair was making progress but the hair lagged behind in the end because he stopped at 1.8 kilometers the tortoise on the other hand walked without stopping and made steady progress isn't it easy to make out who won the race and how graphs help us communicate information visually they enable us to understand quickly trends that would otherwise need lengthy written descriptions as you have studied in mathematics a straight line graph helps in solving a linear equation with two variables in this lesson you will learn about graphical representation of motion at the end of this lesson you will be able to explain how graphs can be used to represent and solve problems related to motion plot various types of distance time graphs calculate the speed of an object at a given instant using the distance time graph plot various types of velocity time graphs calculate displacement using a velocity time graph identify the significance of equations of motion derive the equations of motion and explain the concept of uniform circular motion graphs typically contain both an independent and a dependent variable for example consider the graph shown it plots the runs code by a cricket team in a one day match if you observe the cricket match graph the number of overs bold in a match is independent of how many runs came off and over or how many wickets were taken since the number of overs is an independent quantity it is taken on x-axis similarly we can use line graphs to describe the motion of an object in motion graphs too physical quantities such as distance or velocity show dependence on time while time is an independent quantity therefore time is always taken on the x-axis and the other physical quantities are taken along the y-axis you can create graphs in motion by comparing distance displacement or velocity to time by representing the motion of an object graphically you can infer the nature of motion determine its position at any interval calculate its acceleration displacement velocity etc at any instant and derive equations of motion two of the most commonly used motion graphs are the distance time graph and the velocity time graph a distance time graph represents the change in the position of an object with time in such a graph time is taken along the x-axis and distance is taken along the y-axis the graph is plotted adopting a convenient scale for the distance and time for example you can use a scale of 1 is to 10 on the x axis for time a scale of 1 is to 10 means that each unit on the graph represents 10 seconds let's see how a distance time graph is plotted using some sample data to plot the distance time graph we begin with fixing two axes x and y on a sheet of graph paper on the x axis we mark the equal time intervals according to a convenient scale say one unit equals 10 seconds again using a convenience scale on the y-axis we mark the y-axis at equal intervals for example as mentioned earlier your scale could be one unit equals 10 meters for each set of figures locate and plot a point at which the perpendiculars drawn from the respective time and distance values on the axis intersect finally we connect all the points starting from the origin using straight lines this gives us a distance time graph for the data provided considering the nature of the motion of a body a distance time graph may show a body at rest a body with uniform motion and a body with non-uniform motion if a body is at rest and does not move we say the distance traveled by the body is zero no matter how much time has elapsed the distance time graph for such a body at rest is plotted as a straight line along the x-axis however consider a situation where the body has already traveled a distance of 5 meters when we start analyzing data on its movement it means that the body is at five meters at zero seconds if the body continues to be at the same position for the rest of the time then the graph is a straight line parallel to the x-axis when a body exhibits uniform motion it travels equal distances in equal intervals of time for example consider a body moving with uniform speed the data provided represents the position of the body at different instances of time the initial position of the body is zero the graph for this body is a straight line making an angle with the x-axis and passing through the origin thus the graph of a body in uniform motion is an inclined straight line passing through the origin when a body exhibits non-uniform motion it travels either unequal distances in equal intervals of time or equal distances in unequal intervals of time for example the table gives the position of a body at different instances of time when we plot the distance time graph for the data given we observe that the distance time graph for a body having non-uniform motion is a curve let's analyze this data further during the first 10 seconds the distance travelled by the body is 5 meters thus the speed of the body for the first 10 seconds is 0.5 meter per second for the next 10 seconds the distance covered by the body is 10 meters hence the speed of the body during the second 10 second interval is one meter per second we observe that the speed of the body increased with time at a constant rate hence we can say that the body moves with uniform acceleration thus for a body with uniformly accelerated motion the distance time graph is in shape of a parabola now let us consider a body moving unequal distances in equal time intervals as you can see a graph plotted based on the data provided again shows occur now let's look at how we can use the distance time graph to make deductions about speed to determine the speed of an object consider a segment pq of the distance time graph of an object in uniform motion draw a line parallel to the y-axis from point q and another line parallel to the x axis from point p these parallel lines intersect each other at point r to form a triangle pqr on the graph pr denotes the time interval while qr corresponds to the distance you can see from the graph that as the object moves in time interval t2 minus t1 the distance it covers is s2 minus s1 by definition speed is the ratio of distance covered to the time taken in covering that distance therefore the speed of an object v is equal to s2 minus s1 divided by t2 minus t1 note you can also use the term uniform velocity in place of uniform speed if the magnitude of displacement equals the distance traveled by the object along the y axis like the distance time graph the variation in velocity with time for an object moving in a straight line can be represented by a velocity time graph in a velocity time graph time is taken on the x axis while velocity is taken along the y axis using a velocity time graph you can determine the nature of the motion the velocity of the body at any instant displacement of the body as the area under the graph the instantaneous acceleration and the equations of motion along a straight line let's consider the different cases that can be represented by a velocity time graph body at rest body with uniform velocity body with uniform acceleration body with non-uniform acceleration and body with uniform retardation the displacement is zero for a body at rest and hence the velocity is also zero a velocity time graph plotted for a body at rest displays a straight line along the x-axis when a body moves with uniform velocity there is no change in the velocity with respect to time therefore the velocity time graph for a body with uniform velocity will be a straight line parallel to the x-axis the point where the line intercepts the y axis shows the magnitude of the uniform velocity of the body when a body exhibits uniformly accelerated motion the velocity time graph represents a straight line passing through the origin and subtending an angle with the x-axis this is because in this motion the velocity of the body increases uniformly with time the velocity time graph for a body moving with non-uniform acceleration is a curve and not a straight line the velocity time graph for a body moving with uniform retardation is a straight line making an obtuse angle with the x-axis just as you can use a distance time graph to determine the speed of an object you can use a velocity time graph to determine the displacement of an object the area under a velocity time graph gives you the displacement of the body in a given time consider the velocity time graph of a car moving with uniform velocity say 30 meters per second to determine the displacement of the car between points a and b would represent the position of the car at durations 20 seconds and 30 seconds respectively draw perpendicular lines from a and b to intersect the x axis at point c and d respectively the time taken t by the car to move from the position represented by point a to the position represented by the point b is the difference of time t2 and t1 we can say a b is equal to cd is equal to t2 minus t1 is equal to 30 minus 20 is equal to 10 seconds since the car is moving with a uniform velocity the magnitude of its velocity is the same at all points on the graph velocity v is equal to ac is equal to bd is equal to 30 meters per second by definition the velocity of an object is the ratio of displacement to time therefore we can say that displacement is the product of velocity and time taken substituting v and t in the expression s is equal to vt we get s is equal to ac multiplied by a b this is nothing but the area enclosed in rectangle abcd under the velocity time graph thus in this case the displacement of the card is equal to 30 meters per second multiplied by 10 seconds is equal to 300 meters similarly you can also determine the displacement of a body moving with uniform acceleration consider the velocity time graph to determine the displacement of the body between points a and e from both points a and e draw two lines parallel to the y-axis intersecting the x-axis at points b and c respectively from point a draw a line parallel to the x axis to intersect the line c e at d the area under the velocity time graph between points a and e is the sum of the barriers of the rectangle abcd and the triangle ade [Music] hence the displacement is equal to s is equal to a b multiplied by bc plus half of a d multiplied by d e analyzing a velocity time graph enables you to derive the equations that define the relationship between velocity and acceleration of a body moving in a straight line these equations are collectively referred to as equations of motion there are three equations of motion v is equal to u plus a t s is equal to u t plus half a t square 2 a s is equal to v square minus u square these equations are used to determine the position velocity and acceleration of an object we can derive these equations graphically the first equation of motion is known as the velocity time relation consider a velocity time graph for a uniformly accelerated moving object as shown in the graph u is the initial velocity of the object represented by the point a v is the final velocity represented by the point b is the time in which velocity changes from u to v from point b draw two lines b c and b e perpendicular to the x axis and the y-axis respectively from point a draw a line a d parallel to the x-axis to intersect b c at d now o a represents the initial velocity u bc represents the final velocity v oc represents the time t and bd represents the change in velocity in time t o a d c being a rectangle oc is equal to a d and hence a d also represents the time t bc is equal to bt plus dc which is equal to bd plus oa let's call this equation one substituting bc is equal to v and oa is equal to u in equation one we get v is equal to bd plus u or bd is equal to v minus u as equation 2. the acceleration of the object is the ratio of change in velocity to time taken from the graph acceleration a is equal to bd divided by 80 which is equal to bd divided by oc let's call this equation three substituting oc is equal to t in equation three we get is equal to b d divided by t or b d is equal to a t as equation four equating equations two and four we get a t is equal to v minus u therefore v is equal to u plus e the now derive the second equation of motion that is position time relation in the same velocity time graph we used to analyze position time relation let s be the displacement of the object in time t you can determine the displacement of the object as the area of the trapezium oabc under the curve the area of oabc is the sum of the areas of rectangle oadc and triangle abd area of a rectangle is product of its length and breadth while the area of a triangle is half the product of its base and height area of oabc is equal to oa multiplied by 80 plus half of a d multiplied by b d substituting o a is equal to u a d is equal to oc is equal to t and b d is equal to 80 which equals s is equal to u multiplied by t plus half of t multiplied by a t therefore s is equal to u t plus half a t square the third equation of motion called position velocity relation is derived using the first two equations in the velocity-time graph the displacement of the object s is also equal to the area of trapezium oabc area of trapezium oabc is equal to half of oa plus bc multiplied by oc substituting oa is equal to u bc is equal to v and oc is equal to t we get s is equal to half of u plus v multiplied by t as equation 1 rearranging the velocity time relation we get t is equal to v minus u divided by a let's call this equation two substituting equation two in one we get s is equal to v plus u multiplied by v minus u whole divided by 2a by simplifying this expression we get the position velocity relation that is 2 e s is equal to v square minus u square consider the movement of the seconds hand in a clock as the clock ticks the tip of the hand moves in a circular path covering equal distances in equal time intervals in each time interval the hand subtends the same angle at the center of the clock such a motion when an object moves in a circular path with uniform speed subtending equal angles at the center of the circle in equal intervals is called uniform circular motion other examples that you can relate to include the rotation of the moon and the earth a satellite in a circular orbit around the earth and so on how would you calculate the velocity or the distance covered during uniform circular motion let's analyze the example of an athlete who is running along a circular track the distance covered by the athlete at the end of one lap is equal to the circumference of the track s is equal to 2 pi r the speed of an object is the ratio of displacement to time hence the speed of an object in uniform circular motion is v is equal to 2 pi r divided by t where r is the radius of the circular track and t is the time taken to circumnavigate the track once thus you can use this formula to determine the speed of an object in a uniform circular motion this brings us to the end of this lesson on graphical representation of motion in this lesson you learned how to use graphs to analyze trends and derive and apply equations of motion you also learned about uniform circular motion the section on solved problems provides you an opportunity to review some model problems based on these concepts to revisit the key points covered in this lesson please review the flash card at the end of this lesson [Music] [Applause] [Music] [Music] you