{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Use the procedure of integration. If we know the acceleration, then we can find 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 0 v ! = adt 0 t ! = a dt 0 t ! , Hence v - v 0 = at so that v = v 0 + at. Since so that x = x 0 + v 0 t + 1 / 2 at 2 . dx dt = v 0 + at and hence dx x 0 x ! = v 0 dt 0 t ! + atdt 0 t ! From v = v 0 + at; and substituting into x = x 0 + v 0 t + 1 / 2 at 2 And where s = x - x 0 . t = v ! v 0 a x ! x 0 = v 0 (v ! v 0 ) a + 1 2 a (v ! v 0 ) 2 a 2 (x ! x 0 )a = v 0 v ! v 0 2 + 1 2 (v 2 ! 2vv 0 + v 0 2 ) = 1 2 (v 2 ! v 0 2 ) v 2 = v 0 2 + 2as, v = Lim ! x ! t = dx dt , ! t -> 0
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Background image of page 2
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 Galileo’s 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. v 0 = 0 m/s a = g downwards - we chose to take the the +ve direction as down.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}