MOTION IN ONE DIMENSION

Transcription

1 MOTION IN ONE DIMENSION Particle and Body : In mechanics the word particle is used in a very restricted sense as one having negligible dimensions. But a body possesses dimensions and can be considered to consist of an infinite number of particles. But very often the word body is used for the particle, with the same restricted sense of particle. Rest and Motion : A body is said to be in rest if it continues to occupy the same position with respect to surroundings for any length of time. A body is said to be in motion if it changes its position with time with respect to its surroundings. The terms motion and rest are relative. A passenger in a bus is at rest with respect to his co-passenger sitting beside him, but in motion with respect to road. Hills, trees and buildings are at rest with respect to the earth but they are in motion with respect to the sun. Displacement and Distance : In everyday language the words distance and displacement are used in the same sense but in physics those two words have different meanings. Distance travelled by a body is the actual length of the path covered by a moving body irrespective of the direction in which the body travels. When a body moves from one position to another, the shortest (straight line) distance between the initial position and final position of the body along with direction is known as its displacement. So displacement is defined as follows : Displacement : Change of position of a body in a specified direction is called displacement. Displacement is a vector where as distance travelled is a scalar. Distance : The change of position of a body without reference to direction is called distance. Speed : Speed of a body is the distance travelled by it per unit time. It is a scalar, because direction is not specific. Speed = = Velocity : The rate of change of position of a body is called velocity. or, The rate of displacement of a body is called velocity.

2 Velocity = = Velocity is a vector. Both speed and velocity have same units. Generally a car or a train cannot maintain constant speed throughout its journey, therefore the driver can have an idea of the journey only by knowing the average speed. Average speed = Similarly average velocity can be obtained from the formula. Average velocity = The velocity of a particle at a particular instant of its journey is called instantaneous velocity. v = = The velocity of a body is said to be uniform if it has equal displacements in equal intervals of time, however small these intervals may be. A body is said to move with variable velocity if the body covers unequal displacements in equal intervals of time or equal displacements in unequal intervals of time, however small these intervals may be. So, velocity of a body is variable if either its speed is variable or if its direction of motion changes of if both speed and direction change. A body is said to move with uniform speed if it travels equal distances in equal intervals of time, however small these intervals may be. A body is said to move with non-uniform speed, if it travels unequal distances in equal intervals of time, however small these intervals may be. When a train starts from a station, its speed gradually increases from zero to a practically constant maximum speed or when the train gradually decreases till the train comes to rest. When the speed is increasing the speed is decreasing the train is said to be decelerating or retarding. Thus if there is change in the velocity of a body, it is said to possess acceleration. The change may be in the magnitude or direction of the velocity or both. The rate of change of velocity is called acceleration.

3 Acceleration, a = If v is the change in velocity of particle in a short time interval t, then the acceleration a is given by a = a is vector because v is a vector. This is the average acceleration during time t. Its unit is m/s 2. A body is said to move with uniform acceleration if it moves along a straight line with equal changes in velocity in equal intervals of time however small these intervals may be. It has the same units as those of acceleration. A body is said to be moving with variable acceleration if its velocity changes by unequal amounts in equal intervals of time, or by equal amounts in unequal intervals of time. If the velocity of a body gradually decreases its acceleration is negative. The negative acceleration is called retardation. Equation of motion under uniform acceleration : Let u be the initial velocity of a body moving with uniform acceleration a. Let v be the final velocity at the end of time t. The initial velocity of the body = u The final velocity of the body at the end of time t = v. The change of velocity of the body in time t = v u. The rate of change of velocity of the body = According to the definition, the rate of change of velocity is called the acceleration a. a = or v u = at v = u + at Let u be the initial velocity of a body moving with uniform acceleration a. Let v be the final velocity and s the displacement of the body at the end of time t.

4 The initial velocity of the body = u The final velocity of the body at the end of time, t =v. The change of velocity of the body in time, t = v u. The rate of change of velocity or acceleration, a = i.e. v u = at or v = u + at...(1) The average velocity of the body = The distance travelled by the body, s = average velocity time = t (2) From (1) and (2), s = = t s = ut + at 2 Let u be the initial velocity of a body moving with uniform acceleration, a. Let v be the final velocity and s, the displacement the body at the end of time t. The initial velocity of the body = u The final velocity of the body at the end of the time, t = v The change of velocity of the body in time, t = v u The rate of change of velocity or acceleration, = i.e. v u = at (1) The average velocity of the body

5 = The distance travelled by the body, s = average velocity time = t (2) or (u + v) = Multiplying (1) and (2), (v u)(v + u) = at i.e., v 2 - u 2 = 2as v 2 = u 2 + 2as Let u be the initial velocity of a body moving with uniform acceleration a in the same direction. Let S 1 and S 2 be the distances travelled by the body in n and (n 1) seconds respectively. Let S n be the distance travelled by the body in nth second. The distance travelled in n sec is given by S 1 = un + an 2 (1) The distance travelled in (n 1) sec is given by S 2 = u (n 1) + a(n 1) 2 (2) The distance travelled in nth second is given by S n = S 1 S 2 = (un + an 2 ) [u(n 1) + (n 1) 2 ] = un + an 2 un + u - an 2 + an - a S n = u + an - a

6 The graph is a time velocity graph for a body moving with uniform acceleration a. It started with velocity u and gained a velociry v in t seconds. u is represented by A and v is represented by B. v = u + at In time-velocity graph the gradient of the graph gives the acceleration. The gradient of AB is acceleration a of the body. a = = or at = v u v = u + at (i) In time-velocity graph the displacement s of a body is given by the area between the graph and the time axis. In the fig. the area of the trapezium OABD gives the displacement of the body in time t. s = area of trapezium OABD = area of rectangle OACD + area of triangle ABC = (OA OD) + AC BC But slope = acc. a = = acceleration (a)

7 BC = a. AC s = OA OD + AC a AC = u t + t a t s = ut + at 2 (ii) From equations (i) and (ii) v 2 u 2 = 2as Galileo proved that all bodies, light and heavy, fall towards the centre of the earth with the same acceleration when they are allowed to fall freely. The acceleration of a freely falling body due to earth s attraction is called acceleration due to gravity. The equation of motion with constant acceleration are : 1. v = u + at 2. s = ut + at 2 3. v 2 u 2 = 2as 4. S n = u + a(n - ) For a freely falling body dropped from a height, the initial velocity u = 0 and the acceleration a is equal to the acceleration due to gravity g, therefore the equations of motion change as follows : 1. v = gt 2. s = gt 2 3. v 2 u 2 = 2gs 4. S n = u g(n - ) When a body is projected vertically up, the direction of acceleration due to gravity is opposite to the direction of motion of the body, i.e., a = -g. Then, 1. v = u gt 2. s = ut - gt 2 3. v 2 u 2 = -2gs

8 4. s = u g(n - ) Let a body be projected vertically up from the top of a tower of height h with a velocity u. The body reaches the ground in t sec. If we consider the upward direction as positive them the downward direction will be negative. Projected Body The displacement of the body AB = -h. The acceleration of the body is -g and initial velocity is +u. Substituting these values in the equation. -h = ut - gt 2 or h = -ut + gt 2 If u and t are known, the height of the tower can be calculated from the above equation. At the maximum height C the velocity of the body is zero. Because the body is moving vertically up, the accelerations is g, Let X be the height reached by the body above the top of the tower. v 2 u 2 = 2as 0 u 2 = -2gX X = Maximum height reached by the body = h + X = h + u 2 /2g From the above equations and rules we can find (i) maximum height of ascent. (ii) time of ascent and (iii) time of descent etc. The maximum height to which the body rises is given by H = The time of ascent (t) is given by

9 t = The time in which the body falls from the highest point to that from which it was projected upwards,known as time of descent, and is given by t = u/g Velocity of the body on reaching the point from which it was projected upwards is given by v = or v = u The velocity of the body on reaching the point from which it was projected upwards is equal to the velocity with which it was thrown upwards (but in opposite direction). The velocity-time and position-time graphs for uniform motion have the forms shown below : The basic equations for uniform motion are v(t) = v(0) = v; x(t) = x(0) + vt Equivalently :

10 V(t 2 ) = v(t 1 ) = v; x(t 2 ) = x(t 1 ) + v(t 2 t 1 ) If v a and v b are the velocities of two particles a and b as viewed by a stationary observer, we have relative velocity of b as observed by a = v b - v a And relative velocity of a as observed by b = v a - v b Instantaneous speed = = Instantaneous velocity V = Magnitude of instantaneous velocity = Instantaneous speed Average acceleration = Instataneous acceleration = = The uniform acceleration a of a particles is given by a =

11 The velocity-time and position-time graphs of a uniformly accelerated motion have the forms shown on below: The equations for a uniformly accelerated motion are : v(t) = v(0) + at and x(t) = x(0) + v(0)t + at 2 In terms of this notation, the three equations of motion are v = u + at s = ut + at 2 and v 2 u 2 = 2as Properties of Velocity-time and Position-time graphs : (a) The slope of the position-time graph for uniform motion gives the (constant) velocity for this motion. (b) In general, the slope of the position-time graph at any point gives the instantaneous velocity.

12 (c) The slope of the velocity-time of graph for uniformly accelerated motion gives the (constant) acceleration for such a motion. (d) In general, the slope of the velocity-time graph at any point gives the instantaneous acceleration at that point. (e) The area enclosed by the velocity-time graph, the two ordinates at the time instants t 1 and t 2 and the axis gives the distance covered in the time from t 1 and t 2. For a body falling freely under gravity, the acceleration a equals g, the acceleration due to gravity. It is usual to take to the right or upwards displacements, velocities and accelerations as positive. The distance travelled by the particle in the n th second is given by Circular Motion : S n = u + a (n - ) When a body moves in a circular path with uniform speed (constant speed) its motion is said to be circular motion. e.g., a stone tied to a thread rotated in a circular path. When a body moves in a circle with uniform speed, its velocity changes continuously, so that the motion in a circle is accelerated motion as the direction of the motion changes continuously. Radian : The angle of rotation is measured in terms of number of degrees or in term of revolutions of a rotating body. Generally the angular displacement is expressed in radians where an angle in radians is equal to the length of the arc (which substends the angle) divided by the radius of the circle i.e., Angle in radians = =

13 or (in radians) = Hence, = 1 radian when s = r So, the radian is defined as an angle which is subtended at the centre of a circle by an arc having length equal to the radius of the circle. Linear speed and Angular Velocity : Let a body be moving with a uniform speed v in a circular path of radius r. The linear speed v is given by Linear speed = or v = where, v = Linear speed s = Distance travelled t = Time taken Angular velocity : The angular velocity of a body is the angular displacement per unit time and is represented by. Angular velocity = or = = radians/second Here, = Angular velocity

14 = Angular displacement in radians t = time in seconds The angular velocity is expressed as radians per second and can also be written as rad sec -1. Relation Between Linear Speed and Angular Velocity : Linear speed = Angular velocity Radius of the circular path