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ntroduction to Motion in One Dimension A body A certain amount of matter limited in all directions and consequently having a finite size, shape and occupying some definite space is called a body. Particle A particle is defined as a portion of matter infinitesimally small in size so that for the purpose of investigation, the distance between its different parts may be neglected. Thus, a particle has only a definite position, but no dimension. In the problems we are going to discuss, we will consider a body to be a particle for the sake of simplicity. MOTION IN ONE DIMENSION Motion The position of object can change on a straight line (like on x-axis with respect to origin) or on a plane with respect to some fixed point on frame. So we can define motion as follows:- An object or a body is said to be in motion if its position continuously changes with time with reference to a fixed point (or fixed frame of reference). Caution :  The moving object is either a particle, a point object (such as an electron) or an object that moves like a particle. A body is said to be moving like if every portion of it moves in the same direction and at the same rate. Motion in One Dimension When the position of object changes on a straight line i.e. motion of object along straight line is called motion in one dimension. To understand the essential concepts of one dimensional motion we have to go through some basic definitions. Frame of reference One can see the platform from a running train, and it seems that all the objects placed on platform are continuously changing their position. But one, who is on platform, concludes that the objects on the platform are at rest. It means if we will take the trains are reference frame the objects are not stationary and taking reference frame as platform the objects are stationary. So the study of motion is a combined property of the object under study and the observer. Hence there is a need to define a frame of reference under which we have to study the motion of an object. Definition A frame of reference is a set of coordinate axes which is fixed with respect to a space point (a body or an object can also be treated as a point mass therefore it can become a site for fixing a reference frame), which we have arbitrarily chosen as per our observer's requirement. The essential requirement for a frame of reference is that, it should be rigid. Position of an object  The position of an object is defined with respect to some frame of reference. As a convention, we define position of a point (essentially we treat body as a point mass) with the help of three co-ordinates X, Y and Z. Hence X, Y, Z is a set of coordinate axes representing a 3-dimensional space and each point in this space can be uniquely defined with the help of a set of X, Y and Z coordinate, all three axes being mutually perpendicular to each other. The line drawn from origin to the point represents the position vector of that point.   Introduction to Motion in One Dimension Caution:   The equation of motion derived above are possible only in uniformly accelerated motion i.e. the motion in which the acceleration is constant. Illustration:         The nucleus of helium atom (alpha-particle) travels inside a straight hollow tube of length 2.0 meters long which forms part of a particle accelerator. (a) If one assumes uniform acceleration, how long is the particle in the tube if it enters at a speed of 1000 meter/sec and leaves at 9000 meter/sec? (b) What is its acceleration during this interval? Solution:         (a) We choose x-axis parallel to the tube, its positive direction being that in which the particle is moving and its origin at the tube entrance. We are given x and vx and we seek t. The acceleration ax is not involved. Hence we use equation 3, x = x0 + t.         We get                 x = v0 + ½ (vx0) + vx) t, with x0 = 0 or                 t = 2x/(vx0+vx),                 t = ((2)(2.0 meters))/((1000+9000)meters/sec) = 4.0/10-4 sec    Ans.         (b) The acceleration follows from equation 2, vx = vx0 + axt                 => ax = (v0-vx0)/t = ((9000-1000)meters/sec)/(4.0×10(-4) sec)                 = 2.0 × 107 meter/sec2 Ans. Pause :     The above equations of motion are, however, universal and can be derived by using differential calculus as given below:                        Thus, we have derived the same equation of motion using calculus.         To understand the use of calculus in solving the kinematics problems we can look into the following illustrations.  Introduction to Motion in One Dimension Illustration:         The displacement x of a particle moving in one dimension, under the action of a constant force is related to the time t by the equation t = √x + 3 where x is in meter and t is in seconds. Find the displacement of the particle when its velocity is zero. Solution:         Here t = √x + 3 => √x = t - 3         Squaring both sides, we get x = t2 - 6t + 9,         As we know velocity, v = dx/dt         Hence we get v = dx/dt = 2t - 6         Put v = 0, we get, 2t - 6 = 0         .·. t = 3s         When t = 3s, x = t2 - 6t + 9 = 9 - 6(3) + 9 = 0         Hence the displacement of the particle is zero when its velocity is zero. Illustration:         A particle starts from a point whose initial velocity is v1 and it reaches with final velocity v2, at point B which is at a distance 'd' from point A. The path is straight line. If acceleration is proportional to velocity, find the time taken by particle from A to B. Solution:         Here acceleration a is proportional to velocity v.         Hence a α v         => a = kv, where k is constant         => dv/dt = kv ............... (1)         => (dv/ds)(ds/dt) = kv => (dv/ds) v = kv         => dv = k ds         => k = (v2-v1)/d         From equation (1)         => (dv/v) k.dt.dv/v = Kdt = ln v2/v1 = kt         => t = (d/(v2-v1))ln v2/v1                         Ans.