Download ch 2: Motion in 1D

Chapter 2
Motion in One Dimension
Distance, Speed


The average speed
𝑣=
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑡𝑟𝑎𝑣𝑒𝑙𝑒𝑑
𝑡𝑖𝑚𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝑒𝑙𝑎𝑝𝑠𝑒𝑑
=
Δ𝑑
Δ𝑡
Why average? Speed up, slow down, stop…
(board)
Units are …
CT2-1
Quantities in Motion
• Position
• Velocity
• Acceleration
x
v
a
Need reference point for these
e.g. position of car from where?
Position – x of xva

Defined in terms of a frame of
reference



A choice of coordinate axes
1D: x-axis
- ,Origin, +
(board)
Motion diagrams & x vs t
(all do this)
1. Car at rest at origin
2. Car moves to the right at 10 m/s
3. Car at rest speeds up to 10 m/s
to the right
4. Car moving to the right at 10 m/s
slows down to a stop
5. Car speeds up to the right,
cruises, slows to a stop
Motion diagrams & x vs t
1. Car moves to the right at 10 m/s
2. Car moves to the left at 10 m/s
3. Car moves to the left at 20 m/s
Displacement

Defined as the change in position
 Dx  x  x
f
i

f stands for final and i stands for initial

Units are meters (m) in SI

D = ‘final’ – ‘initial’
Average Velocity: v of xva

The average velocity is rate at which
the displacement occurs
vaverage
Dx xf  xi


Dt
tf  ti
Acceleration – a of xva


Changing position  velocity
Changing velocity  acceleration
Dv vf  vi
a

Dt
tf  ti

Units are…
Avg v vs inst v


(board)
avg () = slope of line between A&B, etc
instantaneous () = slope of tangent at A
Instantaneous v, a



(board)
time interval approaches zero Dt 0
v 
lim
Dt  0
Dx
Dt
𝑑𝑥
𝑣=
𝑑𝑡
Similarly, instantaneous acceleration
𝑑𝑣
𝑎=
𝑑𝑡
Derivative  Slope
v = Slope of x-t graph
a = Slope of v-t graph
Average ()  between 2 t’s: t1 & t2
Instantaneous ()  at a given t
Review of slope (+,-,0

Slope = Dy/Dx

v = Dx/Dt
board
)
Let’s practice some…CQs


CT 2-2,2-3,2-4,2-5
CT 2-6, 2-7
This slide intentionally blank
Kinematic Equations

Used in situations with uniform
acceleration
v  vo  at
1
Dx  vt  vo  v  t
2
1 2
Dx  vot  at
2
v 2  vo2  2aDx
Derivation

(board)
Start with definition of ‘a’, calc…
v  vo  at

Start with definition of ‘v’, calc…
1 2
Dx  v o t  at
2
Graphical Interpretation
Interpret Dx=… (board)
Free fall –
only gravity acting

g = 9.80 m/s²


g is always directed downward


When estimating, use g 10 m/s2
Toward the center of the earth
Close to earth’s surface, g constant
Air resistance: we’ll neglect this, why?
Skydiving, baseball, sheet of paper…

Constant accel (contd)



ANOTHER CASE: SKI SLOPE
CT 2-8,2-9,2-10 (motion diagrams, {x,v,a} vs t graphs)
CT 2-11,2-12,2-13, 2-14
Chapter 2, Problem 25


An electric vehicle starts from rest and
accelerates at a rate of 2.4 m/s2 in a
straight line until it reaches a speed of
23 m/s. The vehicle then slows at a
constant rate of 1.2 m/s2 until it stops.
(a) How much time elapses from start
to stop? (b) How far does the vehicle
move from start to stop?
Draw {a,v,x} vs t
Chapter 2, Problem 45

(a) With what speed must a ball be
thrown vertically from ground level to
rise to a maximum height of 27 m? (b)
How long will it be in the air?