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Conclusion
Velocity is not the same at all times. It is changing A changing velocity is called an acceleration.
dt
dva
t
v
t
vva initialfinalaverage
Average Acceleration
Acceleration is the rate of change of the velocity.
Dimensions are L/T2
SI units are m/s²
Example – Constant acceleration a=1 m/s2Time (s) Velocity (m/s)
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
10 10
11 11
12 12
13 13
14 14
15 15
16 16
17 17
18 18
19 19
20 20
0
5
10
15
20
25
0 5 10 15 20 25
time (s)
v (
m/s
)
atvv
vvatvt
va
f
initialf
0
Instantaneous Acceleration
The instantaneous acceleration is the limit of the average acceleration as t approaches 0
2
20lim x x
xt
v dv d xa
t dt dt
Instantaneous Acceleration -- graph The slope of the
velocity vs. time graph is the acceleration
The green line represents the instantaneous acceleration
The blue line is the average acceleration
Acceleration and Velocity, 1
When an object’s velocity and acceleration are in the same direction, the object is speeding up
When an object’s velocity and acceleration are in the opposite direction, the object is slowing down
Example
Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the same
length) Velocity is decreasing (red arrows are getting shorter) Positive velocity and negative acceleration
Some math. Assume constant acceleration
atvv
atvv
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adtdv
adt
dv
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tv
v
t
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f
0
0
00
0 )0(
constant
0
One more calculus trick: the the chain rulechain rule
axvv
vvxxa
vdvdxa
vdvadxdx
dvv
dt
dx
dx
dv
dt
dva
f
f
v
v
x
x
f
2
0for x
)(2
1)(
20
2
0
20
20
00
Kinematic Equations
The kinematic equations may be used to solve any problem involving one-dimensional motion with a constant acceleration
You may need to use two of the equations to solve one problem
Many times there is more than one way to solve a problem
Kinematic Equations, specific
For constant a, Can determine an object’s velocity at any time t
when we know its initial velocity and its acceleration
Does not give any information about displacement
Kinematic Equations, specific
For constant acceleration,
The average velocity can be expressed as the arithmetic mean of the initial and final velocities
2xi xf
x
v vv
Kinematic Equations, specific
For constant acceleration,
Gives final position in terms of velocity and acceleration
Doesn’t tell you about final velocity
21
2f i xi xx x v t a t
Graphical Look at Motion – displacement – time curve The slope of the curve
is the velocity The curved line
indicates the velocity is changing Therefore, there is an
acceleration
Graphical Look at Motion – velocity – time curve The slope gives the
acceleration The straight line
indicates a constant acceleration
The zero slope indicates a constant acceleration
Graphical Look at Motion – acceleration – time curve
Freely Falling Objects
A freely falling object is any object moving freely under the influence of gravity alone.
It does not depend upon the initial motion of the object Dropped – released from rest Thrown downward Thrown upward
Acceleration of Freely Falling Object The acceleration of an object in free fall is directed
downward, regardless of the initial motion The magnitude of free fall acceleration is g = 9.80
m/s2
g decreases with increasing altitude g varies with latitude 9.80 m/s2 is the average at the Earth’s surface
In the English/American system, g=32 ft/sec2.
Acceleration of Free Fall/Rise, cont. We will neglect air resistance Free fall motion is constantly accelerated
motion in one dimension Use the kinematic equations to solve
problems.
Free Fall Example
Let’s drop an object from the topof this building by simply releasing it.
How long will it take to get to the ground?
How fast will it be going when it gets to theground?
What color is the object?
Issues –
Where is the origin?IS y the same thing as x?? How can that be???