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1 Lecture 2 Fundamental concepts of motion Physics is classified either by subject: Mechanics, Thermodynamics, Optics, and Electrodynamics and so on; or by methods; as experimental and theoretical Physics. There are different theories to describe phenomena, like Classical (Newtonian) Physics, Quantum Theory, Statistical Physics, Theory of Special and General Relativity. Classical Mechanics: It investigates the motion of macroscopic bodies or system of bodies traveling at speeds much lower than that of light. The motion of a body is specified if we know the position of the body and their parts in space and time. The position in space is given with respect to a “frame of reference” : that means one fixed point and three fixed directions. Time: Classical Mechanics assumes a universal, unique time to which we are accustomed in everyday life. There is causality in the events; in principle, the position of a body can be determined at every instant of time if all the forces acting on the body now and in the past are given, and the state of the body is known at a certain instant of time. Bodies can change their position, orientation and shape or volume. We can talk about rigid bodies, elastic bodies, and fluids. The motion of rigid bodies is the combination of translation and rotation. If the size of the body is negligible with respect to the extension of its orbit, the body can be modeled with a material point, a particle with zero extension. A material point performs only translational motion. Kinematics of the material point The position of a particle P in the space is specified by its position vector r, pointing from the origin O of the frame of the reference to the particle and with magnitude equal to the distance of the particle from the origin. The position vector is determined by three coordinates. In the Cartesian system of coordinates shown in the picture these coordinates are the projection lengths of r (x,y,z) to the mutually perpendicular coordinate axes. 2 There can be other coordinate systems. The position of an object on the Earth, for example, can be given by longitude, latitude and height above sea level. Degree of freedom means the number of necessary data to specify the position of the body. Sometime the body moves under constraints – on a surface (2 degrees of freedom) or on a line (one degree of freedom). One-dimensional motion: continuous change of position along a specified line A car moves along a road. Its position can be given with the milestones (kilometer-stones) along the road. The function of motion x(t ) is the distance of the body from the reference point “0”along the curve in terms of time “t”. The displacement during the time interval t between t1 and t2 is x x(t 2 ) x(t1 ) . x can be both negative and positive. x The average velocity is defined as displacement per unit time: vav . t The instantaneous velocity is defined as the time derivative of the position x(t t ) x(t ) dx(t ) v(t ) lim . t 0 t dt The speed is the magnitude (absolute value) of the velocity. The path length or distance traveled (s) during a time interval t is the sum (integral) of the absolute value of the elementary displacements. It can be calculated as the average speed multiplied by the length of the time interval . It is always a positive quantity. [lenght ] Unit of speed and velocity: [ speed ] m/ s [time] 1 km/h = (1000 m)/ (3600 s) = 1/3.6 m/s; 1 m/s = (0.001 km)/(1/3600 h)= 0.0013600 km/h = 3.6 km/h. Example The figure shows the time dependence of position x(t) of a car along a road. Determine the displacement and the distance traveled and both the velocity and the speed of the car for each linear segment of the plot. What is the displacement of the car and how far it has reached during the whole time interval between 9-14 h, and how many km-s has it traveled (what is the distance traveled)? What is its average speed during the route? 3 X(t) plot of a car Time interval (h) Displacement Distance traveled Velocity (km/h) Speed (km/h) (km) (km) 9-10 100 100 100 100 10-10.5 0 0 0 0 10.5-11 100 100 200 200 11-12 0 0 0 0 12-14 -300 300 -150 150 The total displacement of the car is -100 km, it is at a 100 km distance from its original position. It traveled 450 km and the average speed is 90 km/h. One-dimensional motion with non-uniform velocity: acceleration and deceleration The acceleration is defined as the time derivative of the velocity: v(t t ) v(t ) dv a(t ) lim . t 0 t dt [ speed ] The unit of acceleration [a] =m/s2 [time] v v2 v1 Uniformly accelerating motion: a const during the whole motion. t t2 t1 The velocity of an uniformly accelerating body is a linear function of time. v(t ) v0 a(t t0 ) Deceleration means negative acceleration. 4 The function of motion x(t) of an uniformly accelerating body can be derived from the area under the v(t) plot. x vavt v(t ) v0 x(t ) x0 (t t0 ) 2 a x(t ) x0 v0 (t t0 ) (t t0 ) 2 . 2 Examples: a. A Formula-1 car reaches its final speed of 216 km/h in 5 seconds. What is its acceleration? How far it has travelled from the starting point when it reached this speed?(Assume uniform acceleration.) 216 km/h = 60 m/s; a = 60/5=12 m/s2; x = a/2 t2 = 652 = 150 m. b. The acceleration of the falling bodies is g=9.81m/s2 here in Hungary. A boy throws a stone upward with v0=3.27 m/s speed from the top of a 19.62 m high building. How long does the stone fall to the ground? What is its velocity at that instant when it reaches the ground? Let us denote the height from the ground by y: y = y0 + v0t - g/2t2 0 = 19.62 + 3.27t + 9.81/2 t2 ; v = v0-gt=5-9.81t1.5 t2 - t - 6 = 0. t = 2.36 s. v=3.27-9.812.36 = - 19.9 m/s. c. A burglar escapes from the police in a car, and drives with 144 km/h. It is 200 m far when the policeman starts his car after him. The maximum speed of the police car is 180 km/h. He must catch the burglar before he reaches the border at 2 km distance. How fast should the policemen accelerate up his car so as to reach the burglar in time? The burglar drives with the constant speed of 40 m/s. We count the time from that instant when the police car starts. The police car accelerates for t1 seconds till it reaches its final 50 m/s speed: a=50/t1. The distance traveled by the burglar is 200 m shorter than the distance traveled by the police. He would reach the border in t=1800/40=45 s. The policeman accelerates his car for t1 seconds, after that he drives with 50 m/s. 2000 m = 25/t1 t12 + 50(45-t1) t1 = 10 s. The policeman has to speed up his car within 10 s. 5 Motion in two or three dimensions The motion of the body is described by a time dependent vector r(t) which means three scalar functions for three coordinates, x(t), y(t), z(t). The displacement in the time interval t t2 t1 is r r(t2 ) r(t1 ) . The velocity and acceleration vectors are defined r (t t ) r (t ) v(t ) lim , t 0 t v(t t ) v(t ) a(t ) lim . t 0 t The speed is the magnitude of the velocity, v vx2 v y2 vz2 . Projectile motion. The two-dimensional motion of the projectile can be resolved into a vertical motion with uniform acceleration equal to g, and a horizontal motion with constant velocity. vx vx 0 v y v y 0 gt x(t ) x0 vx 0t g 2 y (t ) y0 v y 0t t 2 Example: A stone is thrown at 45o angle with 7.07 m/s speed from the top of a 20 m high building. How far from the building will the stone arrive to the ground? (Use the approximate value g = 10m/s2) vx0 = vy0 = 5 m/s 0=20+5t-5t2t=2.56 s x = 5t =12.8 m.