1. MOTION IN A STRAIGHT LINE 1.Branches of PhysicsBranches of Physics 2.Concept of a Point Object, Reference Point and Frame of ReferenceConcept of a Point ObjectReference Point and Frame of Reference 3.Origin of Position and Time; Rest and Motion – Relative TermsOrigin of Position and TimeRest and Motion Relative Terms 4.Motion in One, Two and Three DimensionMotion in One, Two and Three Dimension 5.Motion in a Straight Line – Distance and Displacement, Scalar & VectorMotion in a Straight Line DistanceDisplacementScalar & Vector 6.Speed - Uniform, Variable, Average and Instantaneous SpeedSpeedUniform, Variable, Average Instantaneous Speed 7.Velocity - Uniform, Variable, Average(Graph) and Instantaneous(Graph)VelocityUniform, Variable, AverageGraphInstantaneous(Graph) 8.Difference between Speed and VelocityDifference between Speed and Velocity 9.Uniform Motion in a Straight LineUniform Motion in a Straight Line 10.Position-Time Graph and Velocity-Time Graph of Uniform MotionPosition-Time Graph Velocity-Time Graph 11.Non-uniform Motion – Acceleration (Uniform, Non-Uniform)Non-uniform Motion AccelerationUniformNon-Uniform 12.Position-Time Graph and Velocity-Time Graph of Non-Uniform MotionPosition-Time Graph Velocity-Time Graph 13.Equations of Motion – Normal(1 st,2 nd,3 rd ) / Graphical(1 st,2 nd,3 rd ) / Calculus(1 st,2 nd,3 rd ) Method of DerivationEquations of Motion 1 st2 nd3 rd1 st2 nd3 rd1 st2 nd3 rd 14.Relative Velocity and GraphsRelative Velocity Graphs Created by C. Mani, Education Officer, KVS RO Silchar Next

2. Mechanics Mechanics is a branch of physics that deals with the motion of a body due to the application of force. The two main branches of mechanics are: (a) Statics and (b) Dynamics Statics Statics is the study of the motion of an object under the effect of forces in equilibrium. Dynamics Dynamics is the study of the motion of the objects by taking into account the cause of their change of states (state of rest or motion). Dynamics is classified into (i) Kinematics and (ii) Kinetics Kinematics The study of the motion of the objects without taking into account the cause of their motion is called kinematics. Kinetics Kinetics is the study of motion which relates to the action of forces causing the motion and the mass that is moved. Home Next Previous

3. Concept of a Point Object In mechanics, a particle is a geometrical mass point or a material body of negligible dimensions. It is only a mathematical idealization. Examples: In practice, the nearest approach to a particle is a body, whose size is much smaller than the distance or the length involved. Home Next Previous

4. POSITION, PATH LENGTH AND DISPLACEMENT Reference Point Consider a rectangular coordinate system consisting of three mutually perpendicular axes, labeled X-, Y-, and Z- axes. The point of intersection of these three axes is called origin (O) and serves as the reference point. The coordinates (x, y, z) of an object describe the position of the object with respect to this coordinate system. Frame of reference The coordinate system along with a clock to measure the time constitutes a frame of reference. Positive direction The positive direction of an axis is in the direction of increasing numbers (coordinates). Negative direction The negative direction of an axis is in the direction of decreasing numbers (coordinates). X Z Y O Home Next Previous

5. While describing motion, we use reference point or origin w.r.t. which the motion of other bodies are observed. We can use any object as reference point. For example, a car at rest or in motion can be used as reference point. When you travel in a bus or train you can see the trees, buildings and the poles moving back. To a tree, you are moving forward and to you, the trees are moving back. Both, you and the trees, can serve as reference point but motion can not be described without reference point. What effect do you get when you play video game involving car racing? Home Next Previous

6. 1. The distance measured to the right of the origin of the position axis is taken positive and the distance measured to the left of the origin is taken negative. 2. The origin for position can be shifted to any point on the position axis. 3. The distance between two points on position-axis is not affected due to the shift in the origin of position-axis. Origin, unit and direction of position measurement of an object -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 (m) +X -X Origin, unit and sense of passage of time 1. The time measured to the right of the origin of the time-axis is taken positive and the time measured to the left of the origin is taken negative. 2. The origin of the time-axis can be shifted to any point on the time-axis. 3. The negative time co-ordinate of a point on time-axis means that object reached that point a time that much before the origin of the time-axis i.e. t = 0. 4. The time interval between two points on time-axis is not affected due to the shift in the origin of time-axis. -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 (s) +t -t Home Next Previous

7. When the same point is chosen as origins for position and time: O A B C x = 0 x = 30 km x = 40 km x = 55 km t = 0 t = 6 h t = 8 h t = 11 h When the different points are chosen as origins for position and time: O A B C x = - 40 km x = -10 km x = 0 km x = 15 km t = -6 h t = 0 t = 2 h t = 5 h Origin for position and time Origin for timeOrigin for position Home Next Previous

8. Rest and Motion A ball is at rest w.r.t. a stationary man. A car is at rest w.r.t. a stationary man. A ball is moving w.r.t. a stationary man. A car is moving w.r.t. a stationary man. Home Next Previous

9. Rest A body is said to be at rest if its position remains constant with respect to its surroundings or frame of reference. Examples: Mountains, Buildings, etc. Motion A body is said to be in motion if its position is changing with respect to its surroundings or frame of reference. Examples: 1. Moving cars, buses, trains, cricket ball, etc. 2. All the planets revolving around the Sun 3. Molecules of a gas in motion above 0 K Rest and Motion are relative terms: An object which is at rest can also be in motion simultaneously. Eg. The passengers sitting in a moving train are at rest w.r.t. each other but they are also in motion at the same time w.r.t. the objects like trees, buildings, etc. Home Next Previous

10. Rest and Motion are Relative Terms Car is moving w.r.t. stationary man. Home Next Previous

11. Rest and Motion are Relative Terms Both the cars are at rest w.r.t. stationary man. Both the cars are moving w.r.t. a stationary man. Both the cars are at rest w.r.t. each other. Home Next Previous

12. In the examples of motion of ball and car, man is considered to be at rest (stationary). But, the man is standing on the Earth and the Earth itself moves around the Sun as well as rotates about its own axis. Therefore, man is at rest w.r.t. the Earth but is rotating and revolving around the Sun. That is why rest and motion are relative terms ! Rest and Motion are Relative Terms – How? Home Next Previous

13. A ship is sailing in the ocean. Man-A in the ship is running on the board in the direction opposite to the direction of motion of the ship. Man-B in the ship is standing and watching the Man-A. Analyse the following cases to understand motion and rest ! 1.Man-A w.r.t. Man-B 2.Man-A w.r.t. ship 3.Man-B w.r.t. ship 4.Ship w.r.t. still water 5.Man-A w.r.t. still water 6.Man-B w.r.t. still water 7.Ocean w.r.t. the Earth 8.Ocean w.r.t. the Sun 9.Earth w.r.t. the Sun 10.Ship w.r.t. the Sun 11.The Sun w.r.t. Milky Way Galaxy 12. Milky Way Galaxy w.r.t. other galaxies Your imagination should not ever stop ! Home Next Previous

14. MOTION IN ONE, TWO OR THREE DIMENSIONS One Dimensional Motion The motion of the object is said to be one dimensional if only one of the three coordinates is required to be specified with respect to time. It is also known as rectilinear motion. In such a motion the object moves in a straight line. Example: A train moving in straight track, a man walking in a narrow, leveled road, etc. Two Dimensional Motion The motion of the object is said to be two dimensional if two of the three coordinates are required to be specified with respect to time. In such a motion the object moves in a plane. Example: Ant moving on a floor, a billiard ball moving on a billiard table, etc. Three Dimensional Motion The motion of the object is said to be three dimensional if all the three coordinates are required to be specified with respect to time. Such a motion takes place in space. Example: A flying airplane, bird, kite, etc. Home Next Previous

15. N 5 km 2 km 5 km Distance travelled is 7 km. Distance travelled is 10 km. Path The line joining the successive positions of a moving body is called its path. The length of the actual path between the initial and final position gives the distance travelled by the body. Distance is a scalar. Motion in a Straight Line Illustration Home Next Previous

16. N 5 km 2 km 5 km Displacement is 6.57 km in the direction shown by the arrow mark. Displacement is 0 km. Displacement Displacement is the directed line segment joining the initial and final positions of a moving body. It is a vector. Illustration Home Next Previous

17. If a body changes from one position x 1 to another position x 2, then the displacement Δx in time interval Δt = t 2 – t 1, is Δx = x 2 – x 1 Conclusions about displacement 1.The displacement is a vector quantity. 2.The displacement has units of length. 3.The displacement of an object in a given time interval can be positive, zero or negative. 4.The actual distance travelled by an object in a given time interval can be equal to or greater than the magnitude of the displacement. 5.The displacement of an object between two points does not tell exactly how the object actually moved between those points. 6.The displacement of a particle between two points is a unique path, which can take the particle from its initial to final position. 7.The displacement of an object is not affected due to the shift in the origin of the position-axis. Home Next Previous

18. Scalar Scalar quantity is a physical quantity which has magnitude only. Eg.: Length, Mass, Time, Speed, Energy, etc. Vector Vector quantity is a physical quantity which has both magnitude as well as direction. Eg.: Displacement, Velocity, Acceleration, Momentum, Force, etc. S. No.DistanceDisplacement 1 2 Distance is a scalar quantity.Displacement is a vector quantity. Distance travelled by a moving body cannot be zero. Final displacement of a moving body can be zero. A C B Distance Displacement Home Next Previous

19. Speed The time rate of change of distance of a particle is called speed. Speed = Distance travelled Time taken or v = s t Note: 1.Speed is a scalar quantity. 2.Speed is either positive or zero but never negative. 3.Speed of a running car is measured by ‘speedometer’. 4.Speed is measured in i) cm/s (cm s -1 ) in cgs system of units ii) m/s (m s -1 ) in SI system of units and iii) km/h (km.p.h., km h -1 ) in practical life when distance and time involved are large. Home Next Previous

20. Uniform Speed A particle or a body is said to be moving with uniform speed, if it covers equal distances in equal intervals of time, howsoever small these intervals may be. Variable Speed A particle or a body is said to be moving with variable speed, if it covers unequal distances in equal intervals of time, howsoever small these intervals may be. Average Speed When a body moves with variable speed, the average speed of the body is the ratio of the total distance traveled by it to the total time taken. Average speed = Total distance travelled Total time taken or v av = s tot t tot Home Next Previous

21. Instantaneous Speed When a body is moving with variable speed, the speed of the body at any instant is called instantaneous speed. If a particle covers the 1 st half of the total distance with a speed ‘a’ and the second half with a speed ‘b’, then If a particle covers 1 st 1/3 rd of a distance with a speed ‘a’, 2 nd 1/3 rd of the distance with speed ‘b’ and 3 rd 1/3 rd of the distance with speed ‘c’, then v av = 2ab a + b v av = 3abc ab + bc + ca Position -Time Graph Position x (m) Time t (s) O Position x (m) Time t (s) O Position x (m) Time t (s) O Stationary object An object in uniform motion An object in non-uniform motion Home Next Previous

22. Velocity The time rate of change of displacement of a particle is called velocity. Velocity = Displacement Time taken or v = s t 1 m/s = km/h 18 5 1 km/h = m/s 5 18 Note: 1.Velocity is a vector quantity. 2.Direction of velocity is the same as the direction of displacement of the body. 3.Velocity can be either positive, zero or negative. 4.Velocity can be changed in two ways: i) by changing the speed of the body or ii) by keeping the speed constant but by changing the direction. 5. Velocity is measured in i) cm/s (cm s -1 ) in cgs system of units ii) m/s (m s -1 ) in SI system of units and iii) km/h (km.p.h., km h -1 ) in practical life when distance and time involved are large. Home Next Previous

23. Average Velocity When a body moves with variable velocity, the average velocity of the body is the ratio of the total (net) displacement covered by it to the total time taken. Average velocity = Net displacement Total time taken or v av = s tot t tot For a body moving with uniform acceleration, v av = 2 u + v Average velocity is also defined as the change in position or displacement (Δx) divided by the time intervals (Δt), in which the displacement occurs: v av = x 2 – x 1 t 2 – t 1 or v av = ΔxΔx ΔtΔt Note: No effort or force is required to move the body with uniform velocity. Uniform Velocity A particle or a body is said to be moving with uniform velocity, if it covers equal displacements in equal intervals of time, howsoever small these intervals may be. Variable Velocity A particle or a body is said to be moving with variable velocity, if its speed or its direction or both changes with time. Home Next Previous

24. a) If a particle undergoes a displacement s 1 along a straight line in time t 1 and a displacement s 2 in time t 2 in the same direction, then b) If a particle undergoes a displacement s 1 along a straight line with velocity v 1 and a displacement s 2 with velocity v 2 in the same direction, then c) If a particle travels first half of the displacement along a straight line with velocity v 1 and the next half of the displacement with velocity v 2 in the same direction, then d) If a particle travels for a time t 1 with velocity v 1 and for a time t 2 with velocity v 2 in the same direction, then e) If a particle travels first half of the time with velocity v 1 and the next half of the time with velocity v 2 in the same direction, then v av = s 1 + s 2 t 1 + t 2 v av = (s 1 +s 2 ) v 1 v 2 s 1 v 2 + s 2 v 1 v av = 2 v 1 v 2 v 1 + v 2 v av = V 1 t 2 + v 2 t 2 t 1 + t 2 v av = v 1 + v 2 2 (in the case (b) put s 1 = s 2 ) (in the case (d) put t 1 = t 2 ) Home Next Previous

25. Position x (m) Time t (s) O Position x (m) Time t (s) O Position x (m) Time t (s) O x – t graph for stationary object x – t graph for an object with +ve velocity x – t graph for an object with -ve velocity Position x (m) Time t (s) O The slope of P 1 P 2 gives average velocity. x1x1 t2t2 x2x2 t1t1 P1P1 P2P2 Average Velocity Uniform Velocity Home Next Previous

26. Difference between Speed and Velocity SpeedVelocity 1. Speed is the time rate of change of distance of a body. 1. Velocity is the time rate of change of displacement of a body. 2. Speed tells nothing about the direction of motion of the body. 2. Velocity tells the direction of motion of the body. 4. Speed of the body can be positive or zero. 4. Velocity of the body can be positive, zero or negative. 3. Speed is a scalar quantity.3. Velocity is a vector quantity. 5. Average speed of a moving body can never be zero. 5. Average velocity of a moving body can be zero. Home Next Previous

27. Instantaneous Velocity When a body is moving with variable velocity, the velocity of the body at any instant is called instantaneous velocity. The velocity at an instant is defined as the limit of the average velocity as the time interval Δt becomes infinitesimally small. Suppose we want to calculate the instantaneous velocity at the point P at an instant t. Position x (m) Time t (s) O x1x1 t4t4 x2x2 t3t3 P3P3 P4P4 P1P1 P2P2 t t2t2 t1t1 x4x4 x3x3 P The slope of P 1 P 2 at t 1 and t 2 with intervals of Δt from t, (i.e. t 1 = t- Δt and t 2 = t+ Δt) gives the average velocity at P. The slope of P 3 P 4 at t 3 and t 4 with intervals of Δt/2 from t, (i.e. t 3 = t- Δt/2 and t 4 = t+Δt/2) gives the average velocity at P which is the closer value to the instantaneous velocity. Proceeding this way, Δt may be gradually reduced to approach zero, i.e. Δt → 0 to get the actual value of the instantaneous velocity. Though average speed over a finite interval of time is greater than or equal to the magnitude of the average velocity, instantaneous speed at an instant is equal to the magnitude of the instantaneous velocity at that instant. Why so? v = ΔxΔx ΔtΔt lim Δt→0 dx dt v = or Home Next Previous

28. Uniform Motion in a Straight Line A body is said to be in uniform motion, if it covers equal displacements in equal intervals of time, however small these time intervals may be. Formula for uniform motion At t=0 At t=t 1 At t=t 2 O A B C Suppose the origin of the position axis is point O and the origin for time measurement is taken as the instant, when object is at point A such that OA = x 0. If at time t 1, the object moving with velocity v is at point B such that OB = x 1, then x 1 = x 0 + vt 1 ………..(1) Similarly, if at time t 2, the object is at point C such that OC = x 2, then x 2 = x 0 + vt 2 ………..(2) From equations (1) and (2), x 2 – x 1 = v(t 2 – t 1 ) and x1x1 x2x2 v = x 2 – x 1 t 2 – t 1 x0x0 Home Next Previous

29. The following points are true for Uniform Motion: 1.Generally, the displacement may or may not be equal to the actual distance covered by an object. However, when uniform motion takes place along a straight line in a given direction, the magnitude of the displacement is equal to the actual distance covered by the object. 2.The velocity of uniform motion is same for different choices of t 1 and t 2. 3.The velocity of uniform motion is not affected due to the shift of the origin. 4.The positive value of velocity means object is moving towards right of the origin, while the negative velocity means the motion is towards the left of the origin. 5.For an object to be in uniform motion, no cause or effort, i.e. no force is required. 6.The average and instantaneous velocity in a uniform motion are always equal, as the velocity during uniform motion is same at each point of the path or at each instant. Home Next Previous

30. Position - Time Graph: (Uniform motion) Position ( m ) Time (s) X2X1x0OX2X1x0O t 1 t 2 B A C Slope of the position-time graph gives the velocity of uniform motion. v = slope of AB = BC AC v = x 2 – x 1 t 2 – t 1 or Home Next Previous

31. Velocity -Time Graph: (Uniform motion) Area under velocity-time graph gives the displacement of the body in uniform motion. x 2 – x 1 = area ABCD = v (t 2 – t 1 ) vOvO A B D C Velocity ( m/s ) Time (s) t 1 t 2 Home Next Previous

32. NON-UNIFORM MOTION The particle is said to have non-uniform motion if it covers unequal displacements in equal intervals of time, however small these time intervals may be. Acceleration If the velocity of a body changes either in magnitude or in direction or both, then it is said to have acceleration. For a freely falling body, the velocity changes in magnitude and hence it has acceleration. For a body moving round a circular path with a uniform speed, the velocity changes in direction and hence it has acceleration. For a projectile, whose trajectory is a parabola, the velocity changes in magnitude and in direction, and hence it has acceleration. The acceleration and velocity of a body need not be in the same direction. Eg.: A body thrown vertically upwards. Home Next Previous

33. For a body moving with uniform acceleration, the average velocity is A body can have zero velocity and non-zero acceleration. Eg.: For a particle projected vertically up, velocity at the highest point is zero, but acceleration is -g. If a body has a uniform speed, it may have acceleration. Eg.: Uniform circular motion If a body has uniform velocity, it has no acceleration. When a body moves with uniform acceleration along a straight line and has a distance ‘x’ travelled in the n th second, in the next second it travels a distance x + a, where ‘a’ is the acceleration. Acceleration of free fall in vacuum is uniform and is called acceleration due to gravity (g) and it is equal to 980 cms -2 or 9.8 ms -2. v av = u + v 2 Home Next Previous

34. Acceleration of a particle is defined as the time rate of change of its velocity. or The acceleration of a particle at any instant or at any point is called instantaneous acceleration. or Note: 1.Acceleration is a vector quantity. 2.Direction of acceleration is the same as the direction of velocity of the body. 3.Acceleration can be either positive, zero or negative. 4.Acceleration of a body is zero when it moves with uniform velocity. 5. Acceleration is measured in i) cm/s 2 (cm s -2 ) in cgs system of units ii) m/s 2 (m s -2 ) in SI system of units and iii) km/h 2 (km h -2 ) in practical life when distance and time involved are large. a av = v 2 – v 1 t 2 – t 1 a av = ΔvΔv ΔtΔt a = ΔvΔv ΔtΔt lim Δt→0 dv dt a = d2sd2s dt 2 a = Home Next Previous

35. Eg.2: The motion of a sliding block on a smooth inclined plane is uniformly accelerated motion. Uniform Acceleration A body has uniform acceleration if its velocity changes at a uniform rate. If equal changes of velocity take place in equal intervals of time, however small these intervals may be, then the body is said to be in uniform acceleration. or Eg.1: The motion of a freely falling body is uniformly accelerated motion. Home Next Previous

36. Non-uniform Acceleration A body is said to be moving with non-uniform acceleration, if its velocity increases by unequal amounts in equal intervals of time. A body has non-uniform acceleration if its velocity changes at a non- uniform rate. or Eg.: The motion of a car on a crowded city road. Its speed (velocity) changes continuously. Retardation or Deceleration or Negative Acceleration A body is said to be retarded if its velocity decreases w.r.t. time. A car is decelerating to come to a halt. Home Next Previous

37. Position - Time Graph Uniformly decelerated (Negative acceleration) Position x (m) Time t (s) Position x (m) Time t (s) Uniform motion (Zero acceleration) Position x (m) Time t (s) Uniformly accelerated (Positive acceleration) Home Next Previous

38. Velocity - Time Graph (Uniformly accelerated / decelerated) Velocity (m/s) Time (s) v0v0 v O t Velocity (m/s) Time (s) v0v0 v O t Velocity (m/s) Time (s) -v 0 -v O t t2t2 t1t1 Velocity (m/s) Time (s) v0v0 -v O Motion in positive direction with positive acceleration Motion in positive direction with negative acceleration Motion in negative direction with negative acceleration Motion with negative acceleration. B/n 0 & t 1 in positive x-axis and b/n t 1 & t 2 in negative x-axis Home Next Previous

39. KINEMATIC EQUATIONS OF UNIFORMLY ACCELERATED MOTION Consider a body moving with initial velocity ‘v 0 ’ accelerates at uniform rate ‘a’. Let ‘v’ be the final velocity after time ‘t’ and ‘x’ be the displacement. v0v0 va t We know that: a = v - v 0 t Cross multiplying, v – v 0 = at orv = v 0 + at The equation v = v 0 + at is known as the first equation of motion. First equation of motion Acceleration = Time taken Final velocity - Initial velocity x Home Next Previous

40. we get v av = v 0 + v 2 or The equation x = v 0 t + ½ at 2 is known as the second equation of motion. Second equation of motion Average velocity = 2 Initial velocity + Final velocity Distance travelled = Average velocity x Time From the first equation of motion we have, v = v 0 + at Substituting for v in equation (1), (1) x = (v 0 + v) 2 t x = (v 0 + v 0 + at) 2 t x = (2v 0 + at) 2 t or x = 2v 0 t + at 2 2 or x = v 0 t + ½ at 2 Home Next Previous

41. Third equation of motion From the first equation of motion we have, v – v 0 = at We know that: v av = v 0 + v 2 orv 0 + v = 2v av (1) orv + v 0 = 2v av (2) Multiplying eqns. (1) and (2), we get v 2 - v 0 2 = 2atv av v 2 - v 0 2 = 2ax or v av t = x or v 2 = v 0 2 + 2ax The equation v 2 = v 0 2 + 2ax is known as the third equation of motion. Home Next Previous

42. KINEMATICEQUATIONS OF UNIFORMLY ACCELERATED MOTION BY GRAPHICAL METHOD First equation of motion Acceleration = Time taken for change Change in velocity Velocity (m/s) O C B v v0v0 E AD Time (s) t a = BD AD a = AE OC a = OE - OA OC a = v - v 0 t v – v 0 = at orv = v 0 + at v – v 0 Home Next Previous

43. Second equation of motion The area of trapezium OABC gives the distance travelled. x = ½ x OC x (OA + CB) x = ½ t (v 0 + v) x = ½ t (v 0 + v 0 + at) x = ½ (2v 0 t + at 2 ) x = v 0 t + ½ at 2 Velocity (m/s) Time (s) O C B vE AD t v0v0 Home Next Previous

44. Third equation of motion The area of trapezium OABC gives the distance travelled. x = ½ x OC x (OA + CB) x = ½ x t x (v 0 + v) (v + v 0 ) = 2x t From the first equation of motion we have, (v – v 0 ) = at (2) (1) Multiplying eqns. (1) and (2), we get v 2 - v 0 2 = 2ax or v 2 = v 0 2 + 2ax Velocity (m/s) Time (s) O C B v v0v0 E AD t Home Next Previous

45. Equations of motion of a freely falling body In case of freely falling body, a = g and x = h Therefore, v = v 0 + gt h = v 0 t + ½ gt 2 v 2 = v 0 2 + 2gh If the position co-ordinate is non-zero at t=0, say ‘x 0 ’, then v = v 0 + at x – x 0 = v 0 t + ½ at 2 v 2 = v 0 2 + 2a(x – x 0 ) Home Next Previous

46. Equations of motion (In terms of Calculus) 1) a = dv / dt or dv = a dt  dv =  a dt v0v0 v 0 t  dv = a  dt v0v0 v 0 t (since a is constant (uniform)) v - v 0 = at orv = v 0 + at Integrating both sides, Home Next Previous

47. 2) v = dx / dt dx = v dt But, v = v 0 + at dx = (v 0 + at) dt  dx =  x0x0 x 0 t (v 0 + at) dt  dx =  x0x0 x 0 t v 0 dt + at dt  0 t  dx = v 0  x0x0 x 0 t dt + t dt  0 t a x – x 0 = v 0 t + ½ at 2 Integrating both sides, Home Next Previous

48. 3) a = dv dt a = dv dx x dt a = dv dx v v dv = a dx Integrating both sides,  a dx x0x0 x  v dv = v0v0 v (since a is constant (uniform))  dx x0x0 x  v dv = a v0v0 v ½ (v 2 – v 0 2 ) = a(x – x 0 ) (v 2 – v 0 2 ) = 2a(x – x 0 ) Home Next Previous

49. Relative Velocity Relative velocity of an object A with respect to another object B, is the rate at which object A changes its position with respect to object B. If v A and v B be the velocities of object A and B respectively, then relative velocity of A w.r.t. B is v AB = v A - v B Similarly, relative velocity of B w.r.t. A is v BA = v B - v A Let us consider two objects A and B moving uniformly with velocities v A and v B in one direction. Let x A (0) and x B (0) be the positions of the objects at t = 0 from the origin O. Therefore the positions of two objects after time t will be given by x A (t) = x A (0) + v A t and x B (t) = x B (0) + v B t x B (t) - x A (t) = [x B (0) - x A (0)] + (v B - v A ) t where x B (t) - x A (t) is relative displacement at time t and x B (0) - x A (0) is relative displacement at time t 0. Then, (v B - v A ) is the relative velocity of B w.r.t. A Home Next Previous

50. Special Cases 1) When the two objects move with equal velocities 2) When the two objects move with unequal velocities i) When v A > v B ii) When v A < v B t (s) x B (0) x A (0) x(m) O t (s) x B (0) x A (0) x(m) O t (s) Meeting time Meeting Position x B (0) x A (0) O Home Next Previous

51. Acknowledgement 1. Physics Part I for Class XI by NCERT 2. Body of the car from net 3. Walking man from the net ] Home End Previous