Post on 18-Jan-2018
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PHY 205 Ch2: Motion in 1 dimension
2.1 Displacement Velocity and Speed2.2 Acceleration2.3 Motion with Constant a2.4 Integration
Ch2: Motion in 1 dim.
2.1 Displacement, Velocity and Speed Ch2: Motion in 1 dim.
Position, displacement, average velocity, speed defined:
Choose axis, origin and direction. Assumption 1 dim space =real line
2.1 Displacement Velocity and Speed Ch2: Motion in 1 dim.
Average velocity: graphical definition:
Start with Position versus time graph for motion in ONE DIMENSION:
Then, compute Displacement between tP and tQ:
Then, Avg. velocity between tP and tQ:
Position x
Time t
Graphically avg velocity = slope of secant PQ in x vs t graph
∆ 𝑥=𝑥𝑄−𝑥𝑃
Q PAvg
Q P
x xv
t t
Thus:
2.1 Displacement Velocity and Speed Ch2: Motion in 1 dim.
Instantaneous velocity: graphical and algebraic definitions
vx = Slope of tangent to graph x vs t at tP
Thus, from that, we also get the graphical definition of v as:
2.2 Acceleration Ch2: Motion in 1 dim.
Graphical and algebraic definitions of acceleration (change of velocity with respect to time)
2.3 Motion with constant acceleration Ch2: Motion in 1 dim.
If rusty with calc, just proceed backwards (not elegant but effective!)
First lets assume a position X dependent on time as follows:Where A, B and C are constants)
2( )x t At Bt C
First avg velocity from generic t to t1=t+Δt:2 2
1 1 1
1 1
( ) ( )Avg
x x At Bt C At Bt Cxvt t t t t
2 21 1 1 1 1
1 1
( ) ( ) ( )( ) ( )A t t B t t A t t t t B t tt t t t
1( ) ( ) 2Avgv A t t B A t t t B At B A t
Same idea for acceleration from velocity, we get
( ) 2v t At B So for instantaneous we let Δt ->0 and get:
( ) 2a t A
Notice that constant A is 1/2a abd B=v(t=0) and C=x(t=0)
2.3 Motion with constant acceleration Ch2: Motion in 1 dim.
So that for motion in 1 dim at constant acceleration a, we have the general equations:(these are the fundamental equations) – Make sure you understand meaning of all symbols!
20 0
0
1( )2
( )
x t at v t x
v t at v
From the above equations we can derive (not fundamental) other equations: :
2 20 0( ) 2 ( ( ) )v t v a x t x
Free fall: def. Constant downward acceleration g where g=9.81m/s2
IF we take positive axis upward, and call position “y” then: note that g is NEVER negative – it’s just short hand for 9.81m/s2 Note also: • “top of trajectory” determined by: vy=0• “hits ground” determined by y=0 (if we take origin at
ground level) but not vy=0
20 0
0
1( )2
( )
y t gt v t y
v t gt v
2 1 1 2
2 1
( ) ( )2Avg
x t x t v vvt t
etc….
2.4 Integration Ch2: Motion in 1 dim.
Taking derivatives from x (or antiderivatives from acceleration a ) we get