Chapter 3 MOTION IN A STRAIGTH LINE 1) What is mechanics ? It is a branch of physics which deals with the study of state of rest and state of motion. 2) Mention the branches of mechanics ? Statics and Dynamics 3) What is statics ? It is a branch of mechanics which deals with the particles at state of rest. 4) What is dynamics ? It is a branch of mechanics which deals with the particles in motion 5) What are the branches of dynamics ? Kinematics and Kinetics 6) What is kinematics ? It is a branch of dy namics which deals with the particle in motion without considering the cause for motion. 7) What is kinetics ? It is a branch of dynamics which deals with the particle in motion by considering the cause for motion. 8) What is rest ? If the particle does not change its position with respect to time, then the particle is said to be at rest. 9) Define motion ? The change in the position of an object with time is called Motion. 10) What is rectilinear motion ? Motion of an object along straight line is called rectilinear motion. 11) What is meant by point object? An object may be treated as point object if the size of the object is much smaller than the distance it moves in a reasonable duration of time. 12) What is frame of reference? The position of an object can be identified using a coordinate system. This coordin ate system along with time is called frame of reference. 13) Define distance. Explain it briefly. Total path travelled by the body is called distance. • Distance is a scalar quantity. It has only magnitude and no specified direction. • SI unit of distance is mete r (m) • Dimensional formula of distance is [L] • Distance or Path length is always positive. It is never be zero or negative for a given motion. • Let the car moves from O to P. Then the distance moved by the car is OP=360m. This distance is called the path leng th. • If the car moves from O to P and then moves back from P to Q, during this course of motion, the path length is OP+PQ =360+120=480m. • Path length is either equal or greater than displacement. 14) Define displacement. Discuss briefly. The shortest distance between initial and final position is called Displacement. Or change in a position of an object in a particular direction is called displacement. • It is a vector quantity. It has both magnitude and direction. • SI unit of displacement is meter (m) and dimensional formula is [L] • Displacement may be positive, zero or negative. • Displacement is either equal or lesser than Path length. • If x 1 and x 2 are the positions of an object in time t 1 and t 2 ; then its displacement is Δx = 𝑥 2 − 𝑥 1 in time interval Δt = 𝑡 2 − 𝑡 1 • In the above diagram ; If car moves from O to P then displacement Δx = 𝑥 2 − 𝑥 1 = 360 - 0 = 360m • If car moves from P to Q; Δx = 240 - 360 = - 120m • If car moves from O to P and back to Q then Δx = 240 - 0 = 240m • If the magnitude of displacement is zero then path length is not zero. • When a particle completes a circular path of radius r, then its displacement is zero but path length is 2πr. • If the car starts from O, goes to P and then returns to O. The final position coinc ides with the initial position. Here displacement is zero but path length is 360+360 = 720m. • distance | displacement | ≥ 1 • In motion along a straight - line distance = |displacement| • In all other cases distance ≥ |displacement| 15) Define speed It is distance travelled by the body per unit time. speed = distance time or 𝑣 = 𝑠 𝑡 • Speed is a scalar quantity and its SI unit is ms - 1 • Dimensional formula of speed is [LT - 1 ] 16) Mention the types of speed • Uniform speed • Non - uniform speed • Average speed • Instantaneous speed 17) Define uniform speed. An object is said to be moving with a uniform speed if it covers equal distance in equal interval of time. 18) Define non - uniform speed (variable speed) An object is said to be moving with non - uniform speed if it cov ers equal distance in unequal interval of time or unequal distance in equal interval of time or unequal distance in unequal intervals of time. 19) Define average speed It is defined as the total distance travelled by the object to the total time taken. Averag e speed v ̅ = total distance total time v ̅ = s 1 + s 2 + s 3 + ... ... ... ... ... t 1 + t 2 + t 3 + ... ... ... ... ... ... NOTE: • If the body covers first half distance with speed v 1 and next half distance with speed v 2 then, Average speed or velocity 𝑣 ̅ = 2 𝑣 1 𝑣 2 𝑣 1 + 𝑣 2 • If a body covers 1 st one third distance at speed v 1 , next one third at speed v 2 and last one third dis- tance with speed v 3 then 𝑣 ̅ = 3 𝑣 1 𝑣 2 𝑣 3 𝑣 1 𝑣 2 + 𝑣 2 𝑣 3 + 𝑣 3 𝑣 1 • If a body travel with uniform speed v 1 for time t 1 and with uniform speed v 2 for time t 2 then average velocity 𝑣 ̅ = 𝑣 1 𝑡 1 + 𝑣 2 𝑡 2 𝑡 1 + 𝑡 2 • If a body travels with uniform speed v 1 for time t and with uniform speed v 2 for another time t then; 𝑣 ̅ = 𝑣 1 + 𝑣 2 2 20) Define instantaneous s peed It is the speed of an object at any particular instant of time. 𝑣 𝑖𝑛𝑠𝑡 = lim ∆ 𝑡 → 0 ∆ 𝑥 ∆ 𝑡 = 𝑑𝑥 𝑑𝑡 21) Define velocity ? It is defined as the rate of displacement with respect to time. velocity = displacement time • Velocity is vector quantity and its SI unit is ms - 1 • Dimensional formula of velocity is [LT - 1 ] 22) Mention the types of velocity. • Uniform velocity • Non - uniform velocity or variable velocity • Average velocity • Instantaneous velocity 23) Define uniform velocity A body is said to be moving with uniform velocity if it undergoes equal displacement in equal intervals of time. 24) Define non - uniform velocity or variable velocity. If an object covers equal displacement in unequal interval of time or unequal displacement in equal interval of time or unequa l displacement in unequal intervals of time, then its velocity is said to be non - uniform velocity. 25) Define average velocity ? [ D K 20 15, D K 20 17 ] It is the ratio of total displacement to the total time taken. 26) Define instantaneous velocity? [ D K 20 17 ] Velocity of a particle at a particular instant of time is called Instantaneous velocity. Instantaneous velocity 𝑣 𝑖𝑛𝑠𝑡 = lim ∆ 𝑡 → 0 ∆ 𝑥 ∆ 𝑡 = 𝑑𝑥 𝑑𝑡 NOTE: • Magnitude of average velocity can be zero but average speed cannot be zero for a moving object. | averag e velocity | average speed ≤ 1 • |Instantaneous velocity| = instantaneous speed • A particle may have constant speed but variable velocity (in circular path). 27) Differentiate between velocity and speed. [ D K 20 16 ] V elocity S peed It is the rate of change of displacement. It is the rate of change of position. It is vector quantity It is scalar quantity. It can be positive, negative or zero for a given motion. It is always positive for a given motion. Magnitude of velocity is always less than or equal to speed. It is always greater than or equal to magnitude of velocity. Problems: 1. The position of an object moving along x - axis is given by x = a + bt 2 where a = 8.5 m, b = 2.5 m s – 2 and t is measured in seconds. What is its velocity at t = 0 s and t = 2.0 s. What is the average velocity between t = 2.0 s and t = 4.0 s? (Ans: v = 15ms - 1 ) 2. A car travels from A to B at a speed of 20 kmph returns to A at a speed of 30 kmph. a) What is the average speed (Ans: v = 6.667 ms - 1 )? b) average velocity for the whole journey. (Ans: v = 0 ms - 1 ) 3. The displacement (in metre) of a particle moving along x - axis is given by 𝑥 = 18 𝑡 + 15 𝑡 2 Calc u late a) instantaneous velocity at t=2s (Ans: v = 78 ms - 1 ) b) average velocity between t=2s and t= 3s (Ans: v = 93 ms - 1 ) c) instantaneous acceleration (Ans: v = 30 ms - 2 ) 4. A man walks on a straight road from his home to a market 2.5 km away with a speed of 5 km h – 1 . Find- ing the market closed, he instantly turns and walks back home with a speed of 7.5 km h – 1 a) What is the magnitud e of average velocity, (Ans: v = 0) 28) Define acceleration. The rate of change of velocity with time is called acceleration. Acceleration = Change in velocity Time taken NOTE: • Acceleration is a vector quantity. • SI unit of acceleration is ms - 2 • Acceleration may be positive, negative or zero • Dimensional formula of acceleration is [M 0 LT - 2 ] • A particle will have acceleration when its speed changes or direction changes or both changes. 29) Define uniform or constant acceleration. If velocity of an object changes by equal amount in every equal time interval, then the acceleration is called uniform or constant acceleration. 30) Define average acceleration. It is defined as the change in velocity to the change in time interval. Average Acceleration = v 2 − v 1 t 2 − t 1 = ∆ v ∆ t 31) Define instantaneous acceleration. [ D K 20 13 ] It is the acceleration of an object at any instant of time and is defined as the rate of change of velocity with respect to time. Instantaneous acceleration 𝑎 𝑖𝑛𝑠𝑡 = lim ∆ 𝑡 → 0 ∆ 𝑣 ∆ 𝑡 = 𝑑𝑣 𝑑𝑡 NOTE: • A particle moving with uniform acceleration from A to B along a straight line has velocity v 1 and v 2 at A and B, then velocity of the particle at mid - point C is equal to v = √ 𝑉 1 2 + 𝑉 2 2 2 • A particle starts from rest and moves with uniform acceleration ‘a’ such that it travels a distance x m in m th second and x n in n th second, then 𝑎 = 𝑋 𝑚 − 𝑋 𝑛 𝑚 − 𝑛 32) What is position - time (x - t) graph. Mention the significance. [ Dharwad 2017] A graph obtained by plotting time along x - axis and position along y - axis is called x - t gra ph. Slope of position - time graph gives velocity. 33) What is velocity - time (v - t) graph. Mention the significance. A graph obtained by plotting time along x - axis and velocity along y - axis is called velocity time graph. Slope of v - t graph gives acceleration. Ar ea under v - t graph gives displacement. 34) What is acceleration - time (a - t) graph. Mention the significance. A graph obtained by plotting time along x - axis and acceleration along y - axis is called a - t graph. Area under a - t graph gives change in velocity. 35) Show that Area under v - t graph gives displacement. Consider an object moving with constant velocity U. Draw v - t graph. Area under v - t graph is the area of the rectangle of height U and base T. Therefore Area = base x height Area = U x T ------- (1) We know that, displacement = velocity × time displacement = 𝑈 × 𝑇 ----------- (2 ) From equation (1) and (2) Area = Displacement Therefore, area under v - t graph gives displacement. 36) Derive kinematic equations for uniform accelerated motion using v - t graph. [D.K 2013, 2014, 2015, Hassan 2016, 2017, Udupi 2016] Consider a body moving in a straight line with uniform acceleration ‘a’. let v 0 be the initial velocity and v is the final velocity after t second. Let x be the displacement v - t graph for the body in shown below. Slope of v - t graph gives acceleration of the particle Slope = acceleration = 𝐵𝐶 𝐴𝐶 𝑎 = 𝑣 − 𝑣 0 𝑡 𝑎𝑡 = 𝑣 − 𝑣 0 .............(1) 𝑣 = 𝑣 0 + 𝑎𝑡 ..............(2) Area under the graph AB gives the displacement X of the body. Displacement X = area of rectangular ACDO +area of triangle ABC X = (OA x OD) + 1 2 (AC x BC) X = v 0 t + 1 2 [ t × ( v − v 0 ) ] (From equation (1)) X = v 0 t + 1 2 𝑎 t 2 ............. (3) We have X = v 0 t + 1 2 at 2 X = v 0 ( v − v 0 a ) + 1 2 a ( v − v 0 a ) 2 X = v 0 ( v − v 0 a ) + 1 2a ( v 2 + v 0 2 − 2v v 0 ) X = 1 2a [ 2 v 0 ( v − v 0 ) + ( v 2 + v 0 2 − 2v v 0 ) ] X = 1 2a ( 2v v 0 − 2 v 0 2 + v 2 + v 0 2 − 2v v 0 ) X = 1 2a ( v 2 − v 0 2 ) 2aX = v 2 − v 0 2 v 2 = v 0 2 + 2aX ..........(4) Equation (2), (3) and (4) are called kinematic equations. NOTE: • If a body starts from rest and moves with uniform acceleration, then ❖ v 0 = 0 ❖ v = at or v α t ❖ X = 1 2 at 2 or X α t 2 ❖ v 2 = 2aX or v 2 α X ❖ X n = 𝑎 2 (2n - 1) or X n α (2n - 1) • Ratio of displacement in I st 1s, I st 2s, I st 3s...............etc is 1:4:9......... ( X α t 2 ) • If a body is projected vertically upward then ❖ v 0 = + v 0 ❖ a = - g ❖ X = +H ❖ t = t ❖ v = +v ❖ At maximum height v = 0 and a = - g ❖ Angle between velocity vector and acceleration vector is 180 0 during upward motion. ❖ At maximum height 𝐻 𝑚𝑎𝑥 = 𝑣 0 2 2 𝑔 ❖ Time of ascent t a = 𝑣 0 𝑔 ❖ In the absence of air resistance t a = t d = 𝑣 0 𝑔 ❖ In the presence of air resistance t a < t d ❖ Time of flight T F = t a + t d = 2 𝑣 0 𝑔 ❖ Irrespective of velocity of projection all the bodies pass through a height 𝑔 2 in the last second of as- cent. ❖ The height reached in the 1 st second of ascent is equal to the height of fall in the last second of de- scent. • When a body is dropped from the height, then ❖ v 0 = 0 ❖ X = - H ❖ v = - v ❖ a = - g ❖ t = t • Galileo’s law of odd number: the distance traversed by a body falling from rest during equal intervals of time are in the ratio of 1:3:5:7....... Since x α (2n - 1) • Distance travelled by a freely falling body in 1 st second is always half of the numerical value o f g i.e. 𝑔 2 = 4 9 𝑚 irrespective of height. • In the absence of air resistance, the velocity of projection is equal to the velocity with which the body strikes the ground. • An object moving under gravity in which air resistance and small change in g are neg lected. • Two bodies are dropped from height H 1 and H 2 simultaneously then after any time the distance be- tween them is equal to H 2 ⁓ H 1 37) What is meant by stopping distance of a vehicle? When brakes are applied to a moving vehicle the distance it travels befo re stopping is called stopping dis- tance. X = 𝑣 0 2 2 𝑎 Problems: 1. A ball is thrown vertically upwards with a velocity of 20 m s – 1 from the top of a multistorey building. The height of the point from where the ball is thrown is 25.0 m from the ground. (a) How high will the ball rise? and (b) how long will it be before the ball hits the ground? Take g = 10 m s – 2 (Ans: (a) x = 20m (b ) t = 5s) 2. A ball is dropped from a height of 90 m on a floor. At each collision with the floor, the ball loses one tenth of its speed. Plot the speed - time graph of its motion between t = 0 to 12 s. Graphs of different cases in motion: Particle at rest: Particle moving with uniform velocity: Particle moving with uniform positive acceleration: Particle moving with uniform negative acceleration: Particle thrown vertically upwards from the earth Position - time and velocity - time graph Graph Position - Time Graph Velocity - Time Graph • Body is at rest • Slope=0 • Velocity=0 • Body moving with constant velocity • Slope=0 • Acceleration=0 • Body moving with infinite velocity. • Practically this is not possible • Velocity of the body in- creases • Slope and Acceleration is infinite. • Practically this is not possi- ble • Body moving with constant velocity or uniform velocity • Slope is constant and positive • Body is moving with con- stant acceleration or uni- form acceleration. • Slope is constant and posi- tive • Velocity is increasing uni- formly with time • Slope is positive at A and B • Velocity is positive at A and B • Slope is increasing from A to B so velocity is also increas- ing from A to B. • Slope is increasing • Acceleration is increasing • Change in velocity is in- creasing • Acceleration is positive • Slope is positive at A and B • Velocity is positive at A and B • Slope is decreasing from A to B • Velocity is decreasing from A to B • Slope is decreasing • Acceleration is decreasing • Change in velocity is de- creasing • Acceleration is positive • Body moves towards the ob- server with increasing speed • Slope is negative • Slope at B>Slope at A • Speed decreases • Velocity is positive and de- creasing • Acceleration is negative and increasing. • Slope increases and nega- tive. • Body moves towards the ob- server with decreasing speed • Slope is negative a t A and B • Slope at A>Slope at B • Slope decreases and nega- tive • Acceleration decreases and negative. • Slope is constant and positive • Velocity is positive and con- stant. • Acceleration is zero • Slope is constant and posi- tive • Acceleration is positive and constant • Slope is constant and nega- tive • Velocity is negative and con- stant • Acceleration is zero • Slope is constant and nega- tive • Acceleration is negative and constant Default graphs: Graph Comment Not possible, since the distance covered cannot decreases with the increase in time. Not Possible, since it gives two positions for the particle at the same instant. Not possible, since po sition changes without change in time. Not possible, since particle will have two velocities at a time Not possible, since particle will have two velocities at a time MCQ QUESTIONS 1) Motion of objects along a straight line is also called as A) Linear motion B) Simple motion C) Rectilinear motion D) None 2) The coordinate system along with a clock constitutes A) Coordinate system B) Cartesian system C) Frame of reference D) None 3) The total distance travelled by the object without considering its direction of motion is called as A) Displacement B) Path length C) Both A and B D) None 4) The body moving in a circular path has A) Constant speed, Variable velocity B) Constant speed, Constant velocity C) Variable speed, Variable velocity D) Variable speed, Constant velocity 5) The particle is said to have acceleration when A) Its speed changes B) Its direction changes C) Both A and B D) N either A nor B 6) The branch of mechanics deals with particle in motion A) Statics B) Dynamics C) Both A and B D) None of these 7) If the particle is covering equal distance in equal interval of time then motion of particle is called A) Non uniform motion B) Uniform motion C) Both A and B D) None 8) If a position time graph shows the straight line parallel to X axis then the particle is A) In uniform motion B) In non - uniform motion C) Stationery or at rest D) None 9) If a position time graph shows the straight line passing through origin then the motion of the parti- cle is A) Uniform motion B) Non uniform motion C) Both A and B D) None 10) If the position of the object is changing continuously with time then the motion is called as A) Uniform motion B) Non uniform motion C) Both A and B D) None 11) The SI unit of velocity is A) m/s B) cm/s C) mm/s D) None 12) The slope of the position time graph of an object moving with positive velocity is A) Zero B) Positive C) Negative D) None 13) The slope of the position time graph of an object moving with negative velocity is A) Positive B) Negative C) Zero D) Infinity 14) The slope of the position time graph gives the value of A) Displacement B) Position C) Velocity D) Acceleration 15) The ratio of total path length and the total time interval required is called as A) Average velocity B) Average speed C) Average acceleration D) None 16) Which is the formula for motion in a straight line A) v = v o + at B) v = v o - a C) v o = 2at + v D) v = 2at + v o 17) Wh ich of the following is one - dimensional motion A) Motion of a train running on a track B) Motion of satellite C) Motion of air particle D) Motion of snake 18) In general, speed is ____ than the magnitude of the velocity. A) Equal B) Less C) Greater D) None 19) The change in velocity of the object with respe ct to time is called as A) Changing velocity B) Variable velocity C) Acceleration D) None 20) Velocity may be A) Positive B) Negative C) Zero D) All 21) Acceleration is the ____ quantity. A) Scalar B) Vector C) Both A and B D) None 22) Acceleration may be A) Positive B) Negative C) Zero D) All 23) In case of velocity time graph, the area under the curve represents the _____ over given time inter- val. A) Distance B) Velocity C) Displacement D) None 24) Velocity - time curve for a body projected vertically upwards A) Straight line B) Ellipse C) Parabola D) Hyperbola 25) The dimensions of average acceleration and instantaneous acceleration is A) [L T - 1 ] B) [L T] C) [ L T - 2 ] D) [L] 26) The SI unit of acceleration is A) m/s B) m/s 2 C) m D) None 27) If the object is thrown vertically upwards then velocity at its uppermost point is A) Maximum B) Minimum C) Zero D) None 28) Kinematic equations are true only for motion in which the magnitude and the direction of accelera- tion are ____ during the course of motion. A) Varying B) Constant C) Variable D) None 29) The ration of the numerical values of the average velocity and average speed of body is A) Unity or Less B) Less than unity C) Unity D) Unity or more 30) If the particle is falling under gravity, this acceleration though negative results in A) Increase in speed B) Decrease in speed C) Constant speed D) None 31) If the particle thrown upward, the negative acceleration results in A) Increase in speed B) Decrease in speed C) Both A and B D) None 32) If the velocity of the particle at any instant is zero then it is ____ that it’s acceleration should also be zero A) Necessary B) Not necess ary C) Both A and B D) None 33) The displacement - time graph of a moving object is a straight line. Then, A) Its acceleration may be uniform B) Its velocity may be uniform C) Its acceleration may be variable D) Both its velocity and acceleration may be uniform 34) If the displacement of an object is zero, then what can we say about its distance covered? A) It is negative B) It is must be zero C) It cannot be zero D) It may or may not be zero 35) Which of the following changes when a particle is moving with uniform velocity? A) Speed B) Velocity C) Acceleration D) Position vector 36) The d istance travelled by an object is directly proportional to the time taken. Its acceleration A) Increases B) Decreases C) Becomes zero D) Remains constant 37) The distance travelled by an object is directly proportional to the time taken. Its speed A) Increases B) Decreases C) Becomes zero D) Remains constant 38) A particle is moving with a constant speed along straight - line path. A force is not required to A) change its direction B) increase its speed C) decrease its momentum D) keep it moving with uniform velocity 39) If the velocity - time graph of an object is a straight - line sloping downwards, th e body has A) zero acceleration B) positive acceleration C) constant acceleration D) negative acceleration 40) When a body is dropped from a tower, then there is an increase in its A) Mass B) Velocity C) Acceleration D) potential energy 41) The velocity time graph of motion of an object starting from rest with uniform acceleration is a straight line A) parallel to time axis B) parallel to velocity axis C) passing through origin D) none of the above 42) If the displacement of a given body is found to be directly proportional to the cube of the time elapsed, then the magnitude of the acceleration of the body is A) Zero B) constant but not zero C) increasing with time D) decreasing with time 43) The acceleration of a moving object can be found from A) area under displacement - time graph B) slope of displacement - time graph C) area under velocity - time graph D) slope of velocity - time graph 44) The total vertical distance covered uy a freely falling body in a given time is directly proportional to A) Time B) square of time C) square of acceleration due to gravity D) product of the time and acceleration due to gravity 45) A simple pendulum hangs from the roof of a train. The string is inclined towards the rear of the train. What is the nature of motion of the train? A) Uniform B) Accelerated C) Retarded D) At rest 46) A bucket is placed in the open where the rain is falling vertically. If a wind begins to blow at double the veloci ty of the rain, how will be the rate of filling of the bucket change? A) Remains unchanged B) Doubled C) Halved D) Becomes four times 47) Kgms - 1 can also be written as A) Nm B) Ns C) Ns - 1 D) Js 48) The slope of the velocity – time graph for retarded motion is A) Zero B) Positive C) Negative D) Neutral 49) A car is moving in a spiral starting from the origin with uniform angular velocity. What can be said about the instantaneous velocity? A) It increases with time B) It decreases with time C) It remains constant D) It does not depend on time 50) What will the velocity v/s time graph of a ball falling from a height before hitting the ground look like? A) A straight line with positive slope B) A strai ght line with negative slope C) A straight line with zero slope D) A parabola ANSWER KEYS 1 2 3 4 5 6 7 8 9 10 C C B A C B B C A B 11 12 13 14 15 16 17 18 19 20 A B B C B A A C C D 21 22 23 24 25 26 27 28 29 30 B D C A C B C B A A 31 32 33 34 35 36 37 38 39 40 B B B D D C D D D B 41 42 43 44 45 46 47 48 49 50 C C D B B A B C A A