Average Acceleration in 1d

1
Average Acceleration in 1d

• average acceleration change in velocity/time
• a Dv/Dt
• we say meters per second, per second
• dimensions are L/T2 gt units are m/s/s m/s2
• may be positive or negative quantity
• not a particularly useful concept . . .
• on a graph of v(t), a slope of line connecting
  two points, or can be calculated from v(t) data
  or functional form
• its impossible to dig out a on a graph of x(t)

2
Average acceleration is a slope
Here, its from time 0 to time t, but can be any
pair of times
3
(Instantaneous) Acceleration in 1d

• acceleration a the average acceleration when
  taken over a VERY SHORT time interval
• non-zero a can be detected LITERALLY in your
  gut!!
• it is the slope of the tangent to the graph of
  v(t) AT the time t

• dimensions are L/T2 gt units are m/s2
• magnitude of acceleration has no special meaning
• avoid use of the word deceleration

4
Average and Instantaneous Kinematical quantities
(drop the instantaneous in practice)
5
Three graphs position velocity
acceleration
6

• Three graphs for the vertically tossed object, or
  any uniform acceleration
• position varies quadratically with t (and is
  always positive in this case, but it could become
  negative)
• velocity steadily (linearly) changes with t (in
  this case through zero and on beyond)
• acceleration never changes!! So a a !!
  (always negative in this case)

7
From These v(t) Graphs, Create a(t) and x(t)
Graphs
8
Constant Acceleration in 1d (along x) full story

• subscript o gt initial time 0
• no subscript gt present or final time t
• here, average acceleration (instantaneous)
  acceleration, so

• two ways to get average velocity one way is via
  the definition

• the other way since v(t) is here a linear
  function, the average value of v and vo is the
  add em and divide by 2 thing, so

9
Now insert (2.7) into (2.9)
And, by solving (2.7) for t and inserting into
(2.9) or (2.10)
These are called the BIG FOUR (along with the
self-evident a constant)
10
Graphical Analysis of x(t) for Constant a
tangent on terminology terms are things that
are added or subtracted factors are things that
are multiplied or divided a factor of 10 is an
order of magnitude
11
The BIG FOUR
12
Experimental fact (Galileo) all objects
accelerate at the same rate near the earth
(value a -g) g 9.81 m/s2 ? 10 m/s2 ?32 ft/s2
Drawing is imperfect

• diver drops from 10 meter cliff
• find velocity at impact, and time

• you may find either quantity first
• to find t directly, use (2.10)
• known vo 0 m, and yo 10 m and y 0 m
• (2.10) simplifies to 0 10 ½ (10) t2 so we
  get t2 2 s2 or t 1.4 s (lucky that linear
  term was not present otherwise must solve
  quadratic equation!!)
• use (2.7) to get v (-10) (1.4) -14 m/s
• see book for reverse order!!

this is the time at which all quantities are
needed