3 motion in two & three dimensions displacement, velocity, acceleration case 1: projectile...
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![Page 1: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/1.jpg)
3 Motion in Two & Three Dimensions
• Displacement, velocity, acceleration
• Case 1: Projectile Motion• Case 2: Circular Motion
• Hk: 51, 55, 69, 77, 85, 91.
![Page 2: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/2.jpg)
Position & Displacement Vectors
jyixr ˆˆ
12 rrr
![Page 3: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/3.jpg)
Velocity Vectors
t
rvav
dt
rdv
222 )()()( zyx vvvv
![Page 4: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/4.jpg)
Relative Velocity
• Examples: • people-mover at airport• airplane flying in wind• passing velocity (difference in velocities)• notation used:
velocity “BA” = velocity of B with respect to A
![Page 5: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/5.jpg)
Example:
![Page 6: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/6.jpg)
Acceleration Vectors
t
vaav
dt
vda
![Page 7: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/7.jpg)
Direction of Acceleration
• Direction of a = direction of velocity change (by definition)
• Examples: rounding a corner, bungee jumper, cannonball (Tipler), Projectile (29, 30 below)
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Projectile Motion
• begins when projecting force ends
• ends when object hits something
• gravity alone acts on object
![Page 9: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/9.jpg)
Horizontal V Constant
![Page 10: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/10.jpg)
Two Dimensional Motion (constant acceleration)
tavv xoxx
221 tatvx xox
tavv yoyy
221 tatvy yoy
![Page 11: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/11.jpg)
Range vs. Angle
![Page 12: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/12.jpg)
Example 1: Calculate Range (R)
vo = 6.00m/s o = 30°
xo = 0, yo = 1.6m; x = R, y = 0
smvv ooox /20.530cos00.6cos
smvv oooy /00.330sin00.6sin
![Page 13: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/13.jpg)
Example 1 (cont.)
2
221
221
9.436.1
)8.9(30sin66.1
tt
tt
tatvy yoy
06.139.4 2 tt
Step 1
![Page 14: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/14.jpg)
Quadratic Equation
02 cbxaxa
acbbx
2
42
06.139.4 2 tt
6.1
3
9.4
c
b
a
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a
acbbx
2
42
Example 1 (cont.)
6.1
3
9.4
c
b
a
)9.4(2
)6.1)(9.4(4)3(3 2 t
)9.4(2
353.63t
954.0
342.0
t
t
End of Step 1
![Page 16: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/16.jpg)
Example 1 (cont.)
tvtatvx oxxox 221Step 2
(ax = 0)
mtvx oo 96.4)954.0(30cos6cos
“Range” = 4.96m
End of Example
![Page 17: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/17.jpg)
Circular Motion
• Uniform
• Non-uniform
• Acceleration of Circular Motion
![Page 18: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/18.jpg)
18
Centripetal Acceleration
• Turning is an acceleration toward center of turn-radius and is called Centripetal Acceleration
• Centripetal is left/right direction
• a(centripetal) = v2/r
• (v = speed, r = radius of turn)
• Ex. V = 6m/s, r = 4m. a(centripetal) = 6^2/4 = 9 m/s/s
![Page 19: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/19.jpg)
Tangential Acceleration
• Direction = forward along path (speed increasing)
• Direction = backward along path (speed decreasing)
t
dvat
![Page 20: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/20.jpg)
Total Acceleration
• Total acceleration = tangential + centripetal
• = forward/backward + left/right
• a(total) = dv/dt (F/B) + v2/r (L/R)
• Ex. Accelerating out of a turn; 4.0 m/s/s (F) + 3.0 m/s/s (L)
• a(total) = 5.0 m/s/s
![Page 21: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/21.jpg)
Summary
• Two dimensional velocity, acceleration
• Projectile motion (downward pointing acceleration)
• Circular Motion (acceleration in any direction within plane of motion)
![Page 22: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/22.jpg)
Ex. A Plane has an air speed vpa = 75m/s. The wind has a velocity with respect to the ground of vag = 8 m/s @ 330°. The plane’s path is due North relative to ground. a) Draw a vector diagram showing the relationship between the air speed and the ground speed. b) Find the ground speed and the compass heading of the plane.
(similar situation)
![Page 23: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/23.jpg)
)v,0()330sin8,330cos8()sin75,cos75(
vvv
pg
agpapg
0330cos8cos75 3.95
pgvm/s 70.7330sin83.95sin75
![Page 24: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/24.jpg)
![Page 25: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/25.jpg)
PM Example 2:
vo = 6.00m/s o = 0°
xo = 0, yo = 1.6m; x = R, y = 0
smvv ooox /00.60cos00.6cos
smvv oooy /00sin00.6sin
![Page 26: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/26.jpg)
PM Example 2 (cont.)
2
221
221
9.406.1
)8.9(0sin66.1
t
tt
tatvy yoy
571.09.4
6.1
6.19.4 2
t
t
Step 1
![Page 27: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/27.jpg)
PM Example 2 (cont.)
tvtatvx oxxox 221Step 2
(ax = 0)
mtvx oo 43.3)571.0(0cos6cos
“Range” = 3.43m
End of Step 2
![Page 28: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/28.jpg)
PM Example 2: Speed at Impact
st 571.0
tavv xoxx tavv yoyy
smtvx /6)0(6 sm
vy
/59.5
571.0)8.9()0(
smvvv yx /20.8)59.5()6( 2222
![Page 29: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/29.jpg)
v1
0
1
2
3
4
5
6
0 2 4 6 8 10 12 14
x(m)
y(m
)
1. v1 and v2 are located on trajectory.
1v
2v
va
![Page 30: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/30.jpg)
Q1. Given 1v2v
v
locate these on the trajectory and form v.
0
1
2
3
4
5
6
0 2 4 6 8 10 12 14
x(m)
y(m
)
1v
2v
![Page 31: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/31.jpg)
Velocity in Two Dimensions
• vavg // r
• instantaneous “v” is limit of “vavg” as t 0
t
rvavg
![Page 32: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/32.jpg)
Acceleration in Two Dimensions
t
vaavg
• aavg // v
• instantaneous “a” is limit of “aavg” as t 0
![Page 33: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/33.jpg)
Displacement in Two Dimensions
ro
r
r
orrr
rrr o
![Page 34: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/34.jpg)
v1
0
1
2
3
4
5
6
0 2 4 6 8 10 12 14
x(m)
y(m
)
1. v1 and v2 are located on trajectory.
1v
2v
va
![Page 35: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/35.jpg)
Ex. If v1(0.00s) = 12m/s, +60° and v2(0.65s) = 7.223 @ +33.83°, find aave.
)39.10,00.6())60sin(0.12),60cos(0.12(1 v
)02.4,00.6())83.33sin(223.7),83.33cos(223.7(2 v
smvvv /)37.6,0()39.10,00.6()02.4,00.6(12
ssms
sm
t
va //)8.9,0(
00.065.0
/)37.6,0(
![Page 36: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/36.jpg)
Q1. Given 1v2v
v
locate these on the trajectory and form v.
0
1
2
3
4
5
6
0 2 4 6 8 10 12 14
x(m)
y(m
)
1v
2v
![Page 37: 3 Motion in Two & Three Dimensions Displacement, velocity, acceleration Case 1: Projectile Motion Case 2: Circular Motion Hk: 51, 55, 69, 77, 85, 91](https://reader030.vdocument.in/reader030/viewer/2022032723/56649cfa5503460f949cbca1/html5/thumbnails/37.jpg)
Q2. If v3(1.15s) = 6.06m/s, -8.32° and v4(1.60s) = 7.997, -41.389°, write the coordinate-forms of these vectors and calculate the average acceleration.
)8777.0,00.6())32.8sin(06.6),32.8cos(06.6(3 v
)2877.5,00.6())39.41sin(997.7),39.41cos(997.7(4 v
smvvv /)41.4,0()8777.0,00.6()2877.5,00.6(12
ssms
sm
t
va //)8.9,0(
15.160.1
/)41.4,0(
v3
v4v a