Page 1 : Downloaded from https:// www.studiestoday.com, , , , y) )) MOTION IN ONE DIMENSION, , , , , , , , , , , , , speed and velocity; acceleration and, , graphs. (Non-uniform acceleration exclud, , , , , , ed, velocity, acceleration; graphs of distance-time and speed-time.,, , Syllabus :, , Scalar and vector quantities, distance,, , Equations of uniformly accelerated n with derivations., Scope - Examples of scalar and vector quantities only, rest, , ‘retardation; distance-time and velocity-time graphs; meaning of slope of the, d). Equations to be derived : v = u + at; S = ut + . at;, , = Fu + ve = w2 + 2aS (equation for $,° is not included), Simple numerical problems., , and motion in one dimension, distance and displacement,, , , , , , , , (A) SOME TERMS RELATED TO MOTION, , , , 2.1 SCALAR AND VECTOR QUANTITIES, , The quantities which we can measure are called, the physical quantities. The physical quantities are, classified into the following two broad categories:, , (1) Scalar quantities or scalars, and (2) Vector, quantities or vectors., , (1) Scalar quantities or scalars : These are the, physical quantities which are expressed only by their, magnitude. For example, if we say that the mass of a, body is 5-0 kg, it has a complete meaning and we are, completely expressing the mass of the body. Thus, we, need the following two parameters to express a scalar, quantity completely :, , (i) Unit in which the quantity is being measured,, , and, (ii) The numerical value of the quantity., , Remember that if the scalar is a pure number, (like 1, e2, etc.), it will have no unit., , Examples : Mass, length, time, distance, density,, volume, speed, temperature, potential (gravitational,, magnetic and electric), work, energy, power, pressure,, quantity of heat, specific heat, charge, electric power,, resistance, density, mechanical advantage, frequency,, angle etc., , Scalar quantities can be added, subtracted,, multiplied and divided by the simple arithmetic, methods. Scalar quantity is symbolically written, by its English letter. For example, mass is, represented by the letter m, time by ¢ and speed, by v., , (2) Vector quantities or vectors : These physical, quantities require the magnitude as well as the, direction to express them, then only their meaning is, , 26, , complete. For example, if we say that “displace a, particle from a point by 5 m”, the first question that, will arise, will be “in which direction”? Obviously,, by saying that the displacement is 5 m, its meaning is, incomplete. But if we say that displace the particle, from that point by 5 metre towards east (or in any, other direction), it has a complete meaning. Thus, we, require the following three meters to express a vector, quantity completely :, , (i) Unit,, , (ii) Numerical value of the quantity and, (iii) Direction., , Examples Displacement, velocity,, acceleration, momentum, force, moment of a force, (or torque), impulse, weight, temperature gradient,, electric field, magnetic field, dipole moment, etc., , The numerical value of a vector quantity, alongwith its unit gives us the magnitude of that, quantity. It is always positive. The negative sign, with a vector quantity implies the reverse (or, opposite) direction. Vector quantities follow, different algebra for their addition, subtraction, and multiplication. A vector quantity is generally, written by its English letter bearing an arrow on, it or by the bold English letter. For example,, , velocity is written as ) or v, acceleration by @, or a, force by F or F. Obviously the forces F, and — F are in opposite directions., 2.2. REST AND MOTION, , Every object in the universe is in motion., , Everyday we see bodies moving around us e.g., birds flying, cars and buses moving, people, , Downloaded from https:// www.studiestoday.com

Page 2 : walking, insdQOWNIRGE ed fro, radi:, , Our earth also moves around the sun so every, thing on it is in a state of motion. The sun and, stars are moving around the centre of their galaxy, and the galaxies too are not at rest., , Although nothing is at rest, but we often say, that a stone lying on the ground is at rest because, the stone does not change its position with respect, to us. Similarly, if we are sitting on a railway, platform and look at a tree nearby, we say that the, tree is at rest because it does not change its, position with respect to us. But when we see a, train leaving the station, we say that the train is, in motion because it is continuously changing its, position with respect to us. Thus,, , , , , , , , , , , , , , For a moving body, if the distance travelled in, a certain time interval is much large as compared, to the size of the body, the body can be assumed to, be a point particle. In this chapter, we shall study, the description of motion of a body assuming it to, be a point particle., , One dimensional motion : When a body, moves along a straight line path, its motion is said, to be one dimensional motion. It is also called, motion in a straight line or rectilinear motion. For, example, the motion of a train on a straight track,, a stone falling down vertically, a car moving on a, long and straight road etc., are one dimensional, (or rectilinear) motions. In such a motion,, there is no movement of the body in lateral, direction (i.e., no sideways motion)., , If a body moves on a plane along a curved, path, its motion is two dimensional and if it, moves in space, its motion is three dimensional., In this chapter, we shall consider only the one, dimensional motion., , Representation of one dimensional motion :, The path of one dimensional motion can be, represented by a straight line parallel to the X-axis, if X-axis is taken in the direction of motion. Each, point on the straight line represents the position of, particle at different instants. The position of particle, at any instant t is expressed by specifying the, , 21, Downloaded from https://, , , , (at t= 0) (at t=1s) (att=2s), , yo stu id Rs OG AY AGO barticte moves,, , its x coordinate will change with time t., , Example : The position of a pebble measured, from its starting point, falling freely and, vertically downwards at different instants is given, in the table below :, , , , , , , , , , , , , , , , , , , , ao : 0 10 8 3 4, Fey Dates ele eet, , , , The motion of the pebble can be represented, by choosing a proper scale for x on a straight line, along X-axis as shown in Fig. 2.1. Here X-axis, represents the vertically downward direction., , tt, x=0 5m 20m 45m x=80m, (at t=35) (at t=4), , Fig. 2.1 Representation of one-dimensional motion, , 2.3. DISTANCE AND DISPLACEMENT, Consider a body moving, , from a point A to a point B, along the path shown in, Fig. 2.2. Then total length of, path from A to B is called, the distance moved by the Fig, 2.2 Motion of a, body, while the length of body fromAtoB, straight line AB in direction from A to B (shown, by the dotted line in Fig. 2.2) is called the, displacement of the body., , Distance, , The total length of path through which a body, moves, is called the distance travelled by it., The distance travelled by a body depends on, the path followed by the body., , It is a scalar quantity. It is generally, represented by the letter S., , Unit : The S.I. unit of distance is metre (m) and, C.GS. unit is centimetre (cm)., , , , , , , , , , , , Displacement, , The shortest distance from the initial to the, final position of the body, is the magnitude of, displacement and its direction is from the, initial position to the final position., It is a vector quantity. It is represented by, >, the symbol SUnit : The S.I. unit of displacement is, metre (m) and C.GS. unit is centimetre (cm)., , , , , , , , , , www.studiestoday.com

Page 3 : The, displacement being a vector, is represented by a, straight line with an arrow, using a convenient, scale. The tip of arrow on the straight line, represents the direction of displacement, while the, length of the straight line on proper scale represents, its magnitude., , In Fig. 2.3, the vector PO: represents, 40 m displacement in east direction with scale, 1 cm = 10 m (displacement). Here origin P is the, initial position and terminus Q is the final position, of the body N, , 40 m DISPLACEMENT, , w E, , P Q, ORIGIN TERMINUS s, , (1) The magnitude of displacement is either, equal to or less than the distance. If motion is, along a fixed direction, the magnitude of, displacement is equal to that of distance, but if, motion is along a curve or any zig-zag path, the, magnitude of displacement is always less than, that of distance. The magnitude of displacement, can never be greater than the distance travelled, by the body., , 3: (i) In Fig. 2.2, the body moves, from A to B along a curved path. The distance, travelled by the body is equal to the length of the, curved path AB, but the displacement of the body, is along the straight line AB shown by the dotted, arrow. Obviously the magnitude of displacement is, less than the distance., , (ii) In Fig. 2.4, a boy travels 4 km towards east and, then 3 km towards north. The total distance travelled, , Distance, , . It is the length of the path traversed by the object, in a certain time., , a:, , 2. It is a scalar quantity i.e., it has only the magnitude. | 2., , 3. It depends on the path followed by the object. 35, , 4. It is always positive. 4., , 5. It can be more than or equal to the magnitude 5., of displacement., , 6. It may not be zero even if displacement is zero, 6., , but it can not be zero if displacement is not zero., , 28, , by the boy is OA + AB = 4 km +3 km=7 km,, but the displacement of the boy is OB = 5 km, , in direction OB i.e., 36-9° due north from east., , NORTH, , , , Thus, the magnitude of displacement is the, length of the straight line between the final and, initial positions., , (2) The distance is the length of path travelled, by the body so it is always positive, but, displacement is the shortest length in direction, from initial position to the final position so it can, be positive or negative depending on its direction., , (3) The displacement can be zero even if the, distance is not zero. If a body, after travelling,, comes back to its starting point, the displacement, is zero but the distance travelled is not zero., , (i) When a body is thrown, vertically upwards from a point A on the ground,, after some time it comes back to the same point A,, then the displacement of the body is zero, but the, distance travelled by the body is not zero (it is 2h, if h is the maximum height attained by the body)., , (ii) A body moving in a circular path when, reaches its original position after one round, then, the displacement at the end of one round is zero,, but the distance travelled by it is equal to the, circumference of the circular path (= 2mr if r is, the radius of the circular path)., , Displacement, , It is the distance travelled by the object in a specified, direction in a certain time (i.e,, it is the shortest, distance between the final and initial positions)., , It is a vector quantity i.e., it has both the magnitude, and direction., , It does not depend on the path followed by the, object., , . It can be positive or negative depending on its direction., . Its magnitude can be less than or equal to the distance,, , but can never be greater than the distance., , . It is zero if distance is zero, but it can be zero even if, , distance is not zero.

Page 4 : Downloaded from https:// www.studiestoday.com, , 2.4 SPEED AND VELOCITY, , For a moving body, speed is the quantity by, which we know how fast the body is moving, while, velocity is the quantity by which we know the speed, of the body as well as its direction of motion. By, speed we do not know the direction of motion of, the body., , (1) Speed, , , , It is a scalar quantity. It is generally, represented by the letter u or v., If a body travels a distance S in time ¢, then, , its speed v is, , Unit of distance, , Unit : Unit of speed = Unit of time, , Since S.I. unit of distance is metre (m) and, of time is second (s), so the S.I. unit of speed is, metre per second (m s~) and its C.GS. unit is, centimetre per second (cm s“!)., , Uniform speed : A body is said to be moving, , with uniform speed if it covers equal distances in, equal intervals of time throughout its motion., , Example : The motion of a ball on a, frictionless plane surface is with uniform speed., , Knowing the uniform speed of a body, we, can calculate the distance moved by the body in, a certain interval of time. If a body moves with, a uniform speed v, the distance travelled by it, in time ¢ is given as :, , Non-uniform or variable speed : A body is, said to be moving with non-uniform (or variable), speed if it covers unequal distances in equal, intervals of time., , Examples : The motion of a ball on a rough, surface, the motion of a car in a crowded street,, the motion of a vehicle leaving or approaching a, destination etc., are with non-uniform speed., , In case of bodies moving with non-uniform, , , , , , (2.2), , speed, we specify their instantaneous speed and, the average speed., , Instantaneous speed : When the speed of a, body keeps on changing, its speed at any instant, is measured by finding the ratio of the distance, travelled in a very short time interval to the time, interval. This speed is called the instantaneous, speed. Thus,, , The speedometer of a vehicle measures the, instantaneous speed., , Average speed : The ratio of the total distance, travelled by the body to the total time of journey is, called its average speed. Thus,, , (2.3), , In case of a body moving with uniform speed,, the instantaneous speed and the average speed, are equal (same as the uniform speed)., , (2) Velocity, , , , Thus, the rate of change of displacement of, a body with time is called the velocity. It is, numerically equal to the displacement of the, body in | s., , It is a vector quantity and is ‘koresentea by, , the symbol @ or \’. For velocity, both its, magnitude and direction must be specified. Two, bodies are said to be moving with same velocities, if both of them move with the same speed in the, same direction. On the other hand, if two bodies, move with the same speed but in different, directions or with different speeds in the same, direction, they are said to be moving with different, velocities., , Unit : The unit of velocity is same as the, unit of speed i.e. the S.I. unit of velocity is metre, per second (ms~!) and the C.G.S. unit is, centimetre per second (cm s7!)., , Uniform velocity : If a body travels equal, distances in a particular direction, in equal, , SSS SS aaa, Downloaded from https:// www.studiestoday.com

Page 5 : Downloaded from https:// www.studiestoda, , intervals of time, the body is said to be moving, with a uniform velocity., , Example : The rain drops reach on earth’s, surface falling with uniform velocity*. A body,, once started on a frictionless surface, moves with, uniform velocity., , If a body moving with a uniform velocity vy,, has displacement 3 ina time interval then by, defirition ¥= 3 /t., , «. Displacement [g=ve}, , Non-uniform or variable velocity : The, velocity of a body can be variable either due to, change in its magnitude or in its direction or in, both magnitude and direction. If a body moves, unequal distances in a particular direction in equal, intervals of time or it moves equal distances in equal, intervals of time, but its direction of motion does, not remain the same, then the velocity of the body, is said to be variable (or non-uniform)., , Examples : The motion of a freely falling, body is with variable velocity because although, the direction of motion of the body does not, change, but the speed continuously increases., Similarly, the motion of a body in a circular path, even with uniform speed is with variable velocity, because in a circular path, the direction of motion, of the body continuously, changes with time. In fact,, its velocity changes at a, uniform rate. At any instant, C A, its velocity is along the, tangent to the circular path =v, , vy, B v, , DY, at that point. Fig. 2.5 shows 7; 9.5 Circular, , the direction of velocity v at aa mae, different points A, B, C and but variable velocity, , D of the circular path., , In case of a body moving with non-uniform, velocity, we specify the instantaneous velocity, and the average velocity., , Instantaneous velocity : For a body moving, with variable velocity, the velocity of the body at, any instant is called its instantaneous velocity. It, * Initially as the rain drop starts falling, first its velocity, increases due to force of gravity, but very soon, due to viscosity, (or friction) and upthrust of air, the viscous force and upthrust, balances the force of gravity on the rain drop with the result, , that the net force on the drop becomes zero. Then the drop, falls down with a uniform velocity called the terminal velocity., , V oday.com :, is measured by finding the ratio of the distance, , travelled in a sufficiently small time interval, to, the time interval. It is important to have time, interval small enough so that the direction of, motion does not change during this interval., , Average velocity : If the velocity of a body, moving in a particular direction changes with time,, the ratio of displacement to the time taken in entire, journey is called its average velocity. Thus,, , +(2.5), , Distinction between speed and velocity, , (1) The speed is a scalar quantity, while, velocity is a vector quantity. The speed of, a body at a given time tells us how fast the, body is moving at that time. The same, information is also obtained by its velocity,, but the velocity also tells us the direction, in which the body is moving., , (2) For the motion in a straight line, the, magnitude of velocity is its speed. The, speed is always positive, but velocity is, given positive or negative sign depending, , upon its direction of motion., , The average velocity of a body can be zero,, even if its average speed is not zero., Examples : (i) If a body starts its motion, from a point and comes back to the same, point after a certain time, the displacement is, zero, so the average velocity is also zero, but, the total distance travelled is not zero and, therefore, the average speed is not zero., , (3), , (ii) If a body moves in a circular path and, covers equal distances in equal intervals of, time, the speed is uniform, but due to, continuous change in its direction of motion,, its velocity is variable. The instantaneous, velocity and instantaneous speed are not zero., The displacement for one round is zero and, therefore, the average velocity is also zero,, but the average speed is 2nr/T if r is the, radius of path and T is the time taken in one, round., , Eee oO Saas, Downloaded from https:// www.studiestoday.com