Week1b - Use the procedure of integration. If we know the...

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Unformatted text preview: Use the procedure of integration. If we know the acceleration, then we can Fnd v (t) and x(t) as follows: so that d v = adt and if in time t, the objects velocity has gone from v 0 to v ; a = lim ! v ! t = dv dt ! t -> 0 dv v v ! = adt t ! = a dt t ! , Hence v - v = at so that v = v + at. Since so that x = x + v t + 1 / 2 at 2 . dx dt = v + at and hence dx x x ! = v dt t ! + atdt t ! From v = v + at; and substituting into x = x + v t + 1 / 2 at 2 And where s = x - x . t = v ! v a x ! x = v (v ! v ) a + 1 2 a (v ! v ) 2 a 2 (x ! x )a = v v ! v 2 + 1 2 (v 2 ! 2vv + v 2 ) = 1 2 (v 2 ! v 2 ) v 2 = v 2 + 2as, v = Lim ! x ! t = dx dt , ! t -> 0 Lets look at a real example - Motion under gravity i.e. falling objects. Obvious - from observation 1). Things fall. 2). The further they fall the faster they move i.e. they accelerate. Less obvious 1). Gravitational acceleration is the same for all falling objects. 2). Gravitational acceleration is constant and does not change as the object falls ( at least near the earths surface). i.e. g is constant with time and is independent of velocity. In any particular problem the equations are selected according to the data available. Often you will need to solve two of the equations together. Everyday experience is that rocks fall faster than feathers - but this is not so in a vacuum (e.g. on moons surface). Because of air resistance objects can reach a terminal falling velocity and it is because of air resistance that objects can fall at different rates. Modern day version of Galileos experiment. (Tipler) In the physics that follows we neglect air resistance. For most objects that we will consider, this is a reasonable approximation. The acceleration due to gravity is denoted the symbol g . i.e a = g ! 9.8 m/s 2 at the surface of the earth. Simple example Say we drop a ball from a window....
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This note was uploaded on 08/20/2010 for the course PHYS 141 taught by Professor Xxx during the One '09 term at University of Wollongong, Australia.

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Week1b - Use the procedure of integration. If we know the...

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