# Motion in one DIMENSION

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```					                     acceleration

 v
a
t
 
v  a t
Simplest case a=constant. Equations hold even if Δt large.
Δv =vf -vi
 
v f  v i  at            ti= 0
Example : If a car traveling at 28 m/s is brought to a full
stop 4.0 s after the brakes are applied, find the average
acceleration during braking.

Given: vi = +28 m/s, vf = 0 m/s, and t = 4.0 s.

v 0  28 m/s
aav                7.0 m/s 2

t    4.0 s
If a = const.
      1       
v av  2 (v i  v f )

Not true in general

If a = const. one dimensional motion
Fig. 04.01

Vav gives

same area
Hence same
distance
Constant acceleration
      1                    
v av  2 (v i  v f )        vf  vi  at
        
      v i  ( v i  at )
v av 
2
       1
v av  v i  2 at
Δx= vavt
Δx= = vit+1/2            at 2
ti = 0
Fig. 04.02
Δx=vxΔt=Area

=

viΔt (blue area)+
½ Δt (aΔt) (gold
area)
=

viΔt + ½ aΔt2
a=constant
•     Δx=vavΔt = 1/2(vi+vf)Δt

but        vf = vi + aΔt so
Δt = (vf-vi)/a
Δx = 1/2(vi+vf)Δt = 1/2 (vi+vf) (vf-vi)/a

•      = 1/(2a) (vf2-vi2) = Δx

•              (vf2-vi2) =2aΔx
Plotted vs time
Minimum length of runway
A fully loaded 747 with all engines at full
throttle accelerates at 2.6 m/s2. Its minimum
takeoff speed is 70 m/s. How long will it
require to reach take off speed? What is the
minimum length of a runway for a 747.

vf = vi + aΔt

(vf2-vi2) =2aΔx
Problem

• A car traveling at a speed of 30 m/s. A deer
runs across the road and the driver slams on
the brakes. It takes .75 s to begin applying the
brakes. With the brakes on the car
decelerates at 6 m/s2. How far does the car
travel from the instant the driver sees the
deer until he stops.
Free Fall
• All objects, under the influence of only gravity
fall. (We are neglecting air resistance)
• They all fall with a constant acceleration
(down) of
•                  g = 9.8 m/s2
• The mass of the object doesn’t matter! Heavy
and light objects all fall with the same g
• It doesn’t matter in which direction it is
moving it has an acceleration of g
• Since we normally take y + up free fall is -g
Free Fall

Slide 2-36
• You drop a stone off a cliff and hear it hit the
ground after two seconds. How high is the
cliff?
• This is an how far question
• Δx=1/2 at2
• Substitute the numbers a=9.8 t=2
• Δx=19.6
• How fast is it going when it hits?
Example Problem
Tennis balls are tested by measuring their bounce when
dropped from a height of approximately 2.5 m. What is
the final speed of a ball dropped from this height?

Slide 2-34
Throwing stones (up)
• What happens if you toss a stone straight up?

v(3)
.   v(4) =0. It reaches its highest point when it stops
going up, i.e. when v = 0

v(1)

a is always downward it is g
v(0)
Throwing stones (up)
• What happens when it starts coming down
again?

.   v(4) =0. It reaches its highest point when it stops
v(3)        going up and then begins to fall with a=-g. It
re-traces its path and velocity but down

v(1)

a is always downward it is g
v(0)
Checking Understanding
An arrow is launched vertically upward. It
moves straight up to a maximum height, then
falls to the ground. The trajectory of the arrow
is noted. Which choice below best represents
the arrow’s acceleration at the different points?

A. A  E  B  D; C  0
B. E  D  C  B  A
C. A  B  C  D  E
D. A  B  D  E; C  0

Slide 2-37
An arrow is launched vertically upward. It
moves straight up to a maximum height, then
falls to the ground. The trajectory of the arrow
is noted. Which choice below best represents
the arrow’s acceleration at the different points?

A. A  E  B  D; C  0
B. E  D  C  B  A
C. A  B  C  D  E
D. A  B  D  E; C  0

Slide 2-38
Checking Understanding
An arrow is launched vertically upward. It moves straight up to a
maximum height, then falls to the ground. The trajectory of the arrow is
noted. Which graph best represents the vertical velocity of the arrow as a
function of time? Ignore air resistance; the only force acting is gravity.

Slide 2-39
An arrow is launched vertically upward. It moves straight up to a
maximum height, then falls to the ground. The trajectory of the arrow is
noted. Which graph best represents the vertical velocity of the arrow as a
function of time? Ignore air resistance; the only force acting is gravity.

D.

Slide 2-40
Example: You throw a ball into the air with speed 15.0
m/s; how high does the ball rise?

Given: viy = +15.0 m/s; ay = 9.8 m/s2
y
viy

x

ay
• Also an how far question
• What is vf? vi? What is a? (be careful of signs!)
• You toss a ball straight up with an initial
vi=25m/s. You then become distracted. How
Example : A penny is dropped from the observation deck of the
Empire State Building 369 m above the ground. With what velocity
does it strike the ground? Ignore air resistance. How long will it take
to hit?
y

Given: viy = 0 m/s, ay =  9.8 m/s2, y =
x
ay   369 m

369 m
Unknown: vyf

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