accelleration -acceleration shows how fast velocity changes - acceleration is the “velocity of...
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ACCELLERATION-Acceleration shows how fast velocity changes
-Acceleration is the “velocity of velocity”
dt
dx
t
xv
t
xv
t
0lim
dt
dv
t
va
t
va
t
0lim
Uniform motion
t
xv
0
0
ttt
xxx
00 ttvxx
Uniform acceleration
t
va
0
0
ttt
vvv
00 ttavv
t
x(t)
t
v(t)
v(t)
t0 t
vv0 Δx
2 0 if
2
2
2
2
2
000
2
00
2
0
000
attvxxttt
tatvxx
tatvx
ttavv
tvv
x
Δx
Δt
tΔt
For any motion:
For uniform acceleration:
tvx
2200 vv
v Δt vv
Δx
Velocity Acceleration
dttvxx
dt
dxtv
t
t
0
0 dttavtv
dt
dva
t
t
0
0
Uniform accelerationUniform motion
00
const
ttvxx
v
2
20
000
0000
00
00
ttattvxx
dtttavxdttvxx
ttav vconst at
t
t
t
Base equations for 1D uniform acceleration
ttt
tavv
tatvxx
0 if
2
0
0
2
002 equations
2 quantities can be found
-What to remember?
-How to use?
Two useful equations that can be derived from the base equations
tvv
x
2
.1 0 See how it was derived on previous slide
20
202 vvxxa
20
20000000
2
00
0
222
2
.2
vvvvvvvvvvvvatvatxxa
attvxx
atvv
1. speeds up all the time.
2. slows down all the time.
3. speeds up part of the time and slows down part of the time.
4. moves at a constant velocity.
time
Example: A train car moves along a long straight track. The graph shows the position as a function of time for this train. The graph shows that the train:
Steepness of slope is decreasing
time
posi
tion
Positive Acceleration = a smile
time
posi
tion
Negative Acceleration = a frown
Example: The graph shows position as a function of time for two trains running on parallel tracks. Which of the following is true?
1. At time t0, both trains have the same velocity.
2. Both trains speed up all the time.
3. Both trains have the same velocity at some time before t0.
4. Somewhere on the graph, both trains have the same acceleration.
t0t1
Same slope at t = t1
Posi
tion
Velo
city
Acc
ele
rati
on
v = slope of x(t)dtdx
v
2
2
dtxd
dtdv
a a = slope of v(t) ora = curvature of x(t)
Posi
tion
Velo
city
Acc
ele
rati
on
0x
Change in velocity = area under a(t) curve
dtavt
t 1
0
dtvxt
t 1
0
Displacement = area under v(t) curve
0v
vt0 t1
a
t
Example v(t) from a(t): Draw the velocity vs. time graph that corresponds to the following acceleration vs. time graph. Assume that the velocity at t = 0 is zero.
B C
v
t
v
t
A
v
t
Does your graph look like one of these?
v
t
NB: a < 0 but object is speeding up.
a
t
NB: a > 0 but object is slowing up.
Free fall
-Free fall acceleration: g=9.8m/s2
Using the two base equations:
atvv
attvxx
0
2
00 2
Substitute the following into the base equations:
yx
ga
To derive the following equations:
gtvv
gttvyy
0
2
00 2
Example 1. A particle, a material point, is thrown vertically up. Find the maximum height the particle will reach and the time it will take, if you are given the initial height and the initial velocity.
Given:
g
Vyy
g
Vt
2
20
0max
01
Unknown variables:
? yt
?yy
v
y
max
max
0
0
at
Solution:
g
vyy
g
gv
g
vvyy
g
v tgtv
22
0
20
0max2
200
00max
0110
Answer:
g
vyy
g
vt
2
20
0max
01
0 no! ?
?
?
11
1
vv
y
t
Example 2. A particle, a material point, is thrown vertically up. Find the velocity with which the particle returns to the point from which it was thrown, and the time this flight will take. The initial height and the initial velocity are given.
Given
12 2tt
?t
?v
yy
v
y
2
2
0
0
0
Solution:
020
02
02
2
20
2
20
2
v v g
vgvv
g
v t
gttv
Answer
02
02
2
vv
g
vt
Compare to example 1:
12 2tt
Example 3, Two particles, material points, are thrown vertically up. One particle is thrown before the other. Find the time at which both objects are at the same height, and the height at which the objects’ intersection occurs.
Equations used:
Given:
?h
?t
t
v
3
0
0 Unknown variables:
?
?
?
2
1
3
y
y
t21 yy
Too many variables but
Thus, 3 equations and 3 unknowns.
Solution:
2
2
2
20
002
2
01
2
00
ttgttvy
gttvy
gΔΔΔtvyy
g
VtttgttV
gt
gttgt
gttVtV
gttV
ttgttV
gttV
0030300
20
20
03
23
0030
23
30
203
030
23
30
2
2
222
22
g
vt ttgttv
gt
gttgt
gttvtv
gttv
ttgttv
gttv
0030300
20
20
03
23
0030
23
30
203
030
23
30
22
222
22
2
2000
20
2000
2
00000
31
2282
222
g
gv
g
vgtgt
g
vtv
g
vtg
g
vtv
tyh
822
20
2000
3
gt
g
v h;
g
vt t Answer:
Free fall (review)
Example1: Ball #1 is thrown vertically upwards with a speed of v0 from the top of a building and hits the ground with speed v1. Ball #2 is thrown vertically downwards from the same place with the same speed v0 and hits the ground with speed v2. Which one of the following three statements is true. Neglect air resistance.A. v1>v2
B. v1=v2
C. v1<v2
D. Depends on which ball is more massiveE. None of the above
Example2: You are throwing a ball straight up in the air. At the highest point, the ball’s
1. velocity and acceleration are zero.
2. velocity is nonzero but its acceleration is zero.
3. acceleration is nonzero, but its velocity is zero.
4. velocity and acceleration are both nonzero.