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