Chapter : D Kinematics Tuesday January 3th Motion in a straight line (D Kinematics) Aerage elocity and aerage speed Instantaneous elocity and speed Acceleration Short summary Constant acceleration a special case Free-fall acceleration Reading: up to page 5 in the text book (Ch. )
Ch.: Motion in one-dimension We will define the position of an object using the ariable x, which measures the position of the object relatie to some reference point (origin) along a straight line (x-axis). Positie direction Negatie direction - -8-6 - - 6 8 x (m)
x (m) 8 6 - - -6-8 - Ch.: Motion in one-dimension Positie direction Negatie direction We will define the position of an object using the ariable x, which measures the position of the object relatie to some reference point (origin) along a straight line (x-axis).
x (m) 8 6 - - -6-8 - Ch.: Motion in one-dimension Positie direction Negatie direction We will define the position of an object using the ariable x, which measures the position of the object relatie to some reference point (origin) along a straight line (x-axis). In general, x will depend on time t. We shall measure x in meters, and t in seconds, i.e. SI units. Although we will only consider only onedimensional motion here, we should not forget that x is a component of a ector. Thus, motion in the +x and -x directions correspond to motions in opposite directions.
Graphing x ersus t 8 6 x(t) x (m) - 3 5 t (s) - -6-8 -
Displacement 8 Displacement Δx: 6 Δx = x - x final - initial position x x (m) - 3 5 t (s) - -6-8 - x Like x, the sign of Δx is crucial Its magnitude represents a distance The sign of Δx specifies direction
x (m) 8 6 - - -6-8 - ag Aerage elocity Δx = = = slopeof line Δt m = =.8m.s 5s 3 5 t (s) Δx = m - (- m) = m Δt = 5 s - = 5 s
Aerage elocity and speed ag Δx x x = = = Δt t t Like displacement, the sign of ag indicates direction Aerage speed s ag : sag = s = total distance Δt s a ag does not specify a direction; it is a scalar as opposed to a ector &, thus, lacks an algebraic sign How do ag and s ag differ?
x (m) 8 6 - - -6-8 - x x ( m) ( m) t t 6s = = = 8 = ms = 6ms 3 3 5 total distance s = Δt 8m+ 8m = = 6ms 6s ms t (s) 8 = ms = 6ms 3
Instantaneous elocity and speed x (m) 8 6 - - -6-8 - Δx dx = lim = = Δ t Δt dt Instantaneous speed = magnitude of 3 local slope 5 3 > > 3 t (s)
Acceleration An object is accelerating if its elocity is changing Aerage acceleration a ag : a ag Δ = a = = Δt t t Instantaneous acceleration a: a Δ = lim = = = Δ t Δt dt dt dt dt d d dx d x This is the second deriatie of the x s. t graph Like x and, acceleration is a ector Note: direction of a need not be the same as
8 6 Decelerating a a x (m) - 3 5 t (s) - -6 x(t) -8 - Accelerating
8 6 Decelerating a a x (m) - (t) 3 5 t (s) - -6 x(t) -8-3 > > Accelerating
8 6 Decelerating a a (m/s) (t) a(t) 3 5 t (s) - Accelerating
Aerage elocity: Summarizing Displacement: Δx = x - x ag Δx x x = = = Δt t t total distance Aerage speed: sag = s = Δt Instantaneous elocity: dx = = local slope of x ersus t graph dt Instantaneous speed: magnitude of
Summarizing Aerage acceleration: a ag Δ = a = = Δt t t Instantaneous acceleration: d a= = local slope of ersus t graph dt In addition: d dx d x a= = =curature of x ersus t graph dt dt dt SI units for a are m/s or m.s - (ft/min also works)
Constant acceleration: a special case a(t) a d Δ = = = dt Δt t a = at or = + at t
Constant acceleration: a special case (t) t () = + at t
Constant acceleration: a special case (t) t () = + at at t
Constant acceleration: a special case (t) t () = + at at Area t = at t
Constant acceleration: a special case (t) t () = + at at Area t = at Area = t t
Constant acceleration: a special case (t) t () = + at Area = t + at Is there any significance to this? t
Constant acceleration: a special case a(t) a Hang on There seems to be a pattern here t = Area = at t
Constant acceleration: a special case It is rigorously true (a mathematical fact): = Area under a(t) cure = at x x = Area under (t) cure = t + / at What we hae discoered here is integration or calculus at () = d dt Δ = d= adt = t Area under cure = at
Constant acceleration: a special case It is rigorously true (a mathematical fact): = Area under a(t) cure = at x x = Area under (t) cure = t + / at What we hae discoered here is integration or calculus t () = dx dt x t t Δ x= dx= () t dt = ( + at) dt=area x x x = t + at
Constant acceleration: a special case x(t) x x = t + at x t
Equations of motion for constant acceleration One can easily eliminate either a, t or o by soling Eqs. -7 and - simultaneously. Equation number.7...9 Equation = + at x x = t + at = + a( x x ) x x = ( + ) t x x = t at Important: equations apply ONLY if acceleration is constant. Missing quantity x x t a
A Real Example: Free fall acceleration If one eliminates the effects of air resistance, one finds that ALL objects accelerate downwards at the same constant rate at the Earth s surface, regardless of their mass (Galileo). That rate is called the free-fall acceleration g. The alue of g aries slightly with latitude, but for this course g is taken to be 9.8 ms - at the earth's surface. It is common to consider y as increasing in the upward direction. Therefore, the acceleration a due to graity is in the negatie y direction, i.e. a y = -g = -9.8 ms -. NOTE: There is nothing special about the parameter y. You can use any labels you like, e.g., x, z, x, etc.. The equations we hae deried work quite generally.
Equations of motion for constant acceleration One can easily eliminate either a y, t or oy by soling Eqs. -7 and - simultaneously. Equation number.7...9 Equation = + a t y y y y y y = t+ a t = + y y y a ( y y ) y y y = ( + y ) t y y y = t a t Important: equations apply ONLY if acceleration is constant. y y Missing quantity y y y t a y y
A Real Example: Free fall acceleration 5 Height y (meters) 3 6 8 6 8 Time t (seconds)
A Real Example: Free fall acceleration Velocity y (m/s) 5-5 6 8 6 8 Time t (seconds) -