Embed Size (px)
Citation preview
1. The x-coordinate of a particle in curvilinear
motion is given by , where x is in meters
and t is in seconds. The y-component of acceleration
in meters per second squared is given by ay=4t. If
the particle has y-components y = 0 and m/s
when t = 0, find the magnitudes of the velocity
and acceleration when t = 2 s. Sketch the path for
the first 2 seconds of motion and show the velocity
and acceleration vectors for t = 2 s.
v
ttx 32 3
4y
a
, ay=4t, y = 0 and m/s when t = 0, find the magnitudes
of the velocity and acceleration when t = 2 s. Sketch the path for
v
ttx 32 3 4y
a
2
23
/24,/212
12,36,32
smasmvstfor
taxvdt
dvtvx
dtdx
ttx
xx
xxx
x
2
3
0
2
0
222
04
/8,/122
43
2
42,
42,22
44
4,4
smasmvstfor
tt
y
dttdyvydt
dy
tvtt
v
dttdvayvdt
dvta
yy
ty
y
yy
tv
yyy
y
y
y
the first 2 seconds of motion and show the velocity and acceleration vectors for t = 2 s.
, ay=4t, y = 0 and m/s when t = 0, find the magnitudes
of the velocity and acceleration when t = 2 s. Sketch the path for
v
ttx 32 3 4y
a
mjtt
ittjyixr
smaaasmjia
smvvvsmjivstfor
jtitjaiaaaa
jtitjvivvvv
yx
yx
yxyx
yxyx
43
232
/3.25,/824
/2.24,/12212
412
4236
33
2222
22
22
the first 2 seconds of motion and show the velocity and acceleration vectors for t = 2 s.
vx 21
12
vy
v
ax 24
8
ay
a
33.13102
25.825.25.1
67.411
08.225.15.0
000
yxt
2. A particle moves in the x-y plane with a y-
component of velocity in meters/second given by
vy=8t with t in seconds. The acceleration of the
particle in the x-direction in meters per second
squared is given by ax=4t with t in seconds. When t=0,
y=2 m, x=0 and vx=0. Find the equation of the path
of the particle and calculate the magnitude of the
velocity of the particle for the instant when its x-
coordinate reaches 18 m.
v
vy=8t m/s , ax=4t m/s2, when t=0, y=2 m, x=0 and vx=0, equation of the path, magnitude of velocity when its x=18 m.
24,42
82
8,
/8,/8
222
02
2
tytt
y
tdtdyvdt
dy
smadt
dvsmtv
ty
y
y
y
y
3
2,2,
22
4,4
4,/4
3
0
2
0
22
00
2
txdttdxv
dt
dx
tt
vtdtdv
tadt
dvsmta
tx
x
x
tv
x
xx
x
x
2
3
12
1
3
2
1
2
122 2
8
1
3
22
2
1
3
2,2
2
1,2
4
1,24
yyxytytyt
taking the square of both sides
3222
322 2144212
yxyx
x direction y direction
equating
vy=8t m/s , ax=4t m/s2, when t=0, y=2 m, x=0 and vx=0, equation of the path, magnitude of velocity when its x=18 m.
smv
vvv
smtv
smtv
stt
xmx
yx
y
x
/30
2418
/24)3(88
/18322
33
218,18
2222
22
3
3. A ball is dropped onto a step at point A and rebounds with a velocity vo at an angle of 15° with the vertical. Determine the value of vo knowing that just before the ball bounces at point B its velocity vB forms an angle of 12° with the vertical. Also determine the velocity of B, vB and the distance d.
d
0.2 m
vo
vB
A
B
15o
12o
Determine the value of vo knowing that just before the ball bounces at point
B its velocity vB forms an angle of 12° with the vertical. Also determine the velocity of B, vB and the distance d.
***905.415cos2.0
)81.9(21
15cos02.0
21
)(
2
0
2
0
2
00
ttv
ttv
gttvyy yB
***15sin,015sin0
21
)(
00
2
00
tvdtvd
tatvxx xxB
d
0.2 m
vo
vB
A
B
15o
12o
x
y
y
x
tvvgtvv Byy 81.915cos12cos)( 00
000 245.115sin12sin)( vvvvvv BBxx
mdsmvsmvst
tttt
tvtvtvv
tvv
B 41.0/324.3/67.2594.0
567.02.0904.415cos)49.4(2.0
49.481.9184.281.9966.0218.1
81.915cos12cos)245.1(
0
22
0000
00
4. The quarterback Q throws the football when the receiver R is in the position shown. The receiver’s velocity is constant at 10 m/s and he catches the ball when it is 2 m above the ground. If the quarterback desires the receiver to catch the ball 2.5 s after the launch instant shown, determine the inital speed v0 and angle q required.
5.2cos055
2
1
0
2
00
qv
tatvxx xx
Ball when caught in y direction
2
0
2
00
5.281.92
15.2sin15.22
2
1
qv
gttvyy y
smvv
v
/17.25,03.29cos
22
cos
22
03.29,555.0tan
tan55506.30,656.30sincos
225.215.0
cos
22
cos5.2
55
00
0
q
qqq
x
y vR=10 m/s=cst , y=2 m, t=2.5 s, v0=?, q =?
Ball when caught in x direction
mtvxx R 55)5.2(10300
5. The x- and y-motions of guides A and B with right
angle slots control the curvilinear motion of the
connecting pin P, which slides in both slots. For a short
interval, the motions are governed by and,
where x and y are in millimeters and t is in
seconds. Calculate the magnitudes of the velocity and
acceleration of the pin for t= 2s. Sketch the direction
of the path and indicate its curvature for this instant.
21420 tx 31615 ty
v
a
Calculate the magnitudes of the velocity and acceleration of
the pin for t= 2s. Sketch the direction of the path and indicate its
curvature for this instant.
v
a
21420 tx
31615 ty
6. A boy tosses a ball onto the roof of a house. For the launch conditions shown, determine the slant distance s to the point of impact. Also, determine the angle q which the velocity of the ball makes with the roof at the moment of impact.
determine the slant distance s to the point of impact. Also, determine the angle q which the velocity of the ball makes with the roof at the moment of impact.
7. A projectile is launched from point A with an initial speed v0 = 30 m/s. Determine the minimum value of the launch angle a for which the projectile will land at point B. Take g = 9.81 m/s2.
A projectile is launched from point A with an initial speed v0 = 30 m/s. Determine the minimum value of the launch angle a for which the projectile will land at point B. Take g = 9.81 m/s2.
aaaaaa
a
a
aa
222
2
2
22
00
00
tan1sec,seccos
1cos30110
905.4cos03110
sin0324
ofvaluetheinputting
905.4sin3002421
//
cos30110
,cos30110,//
t
ttgttvyyy
tttvxxx
y
x
54.3014.47
59.0tan078.1tan
0945.41tan110tan945.65
sec945.65tan11024
cos
130
110905.4tan11024
21
21
2
tan1
2
2
2
2
aa
aa
aa
aa
aa
a x
y
A projectile is launched from point A with an initial speed v0 = 30 m/s. Determine the minimum value of the launch angle a for which the projectile will land at point B. Take g = 9.81 m/s2.
14.47
,
41.6,17.4//14.47
94.2)29.3(905.4)29.3(54.30sin3021
29.3,54.30cos3085//54.30
22
00
00
a
Therefore
myst
mgttvyy
stttvxx
AC
possiblenot
y
ACACx
Check A-C
C
x
y
8. For a certain interval of motion, the pin P is
forced to move in the fixed parabolic slot by the
vertical slotted guide, which moves in the x direction
at the constant rate of 20 mm/s. All measurements
are in mm and s. Calculate the magnitudes of and
of pin P when x = 60 mm.
v
a
vx= 20 mm/s (constant), calculate magnitudes of and of pin P when x = 60 mm.
v
a
9. Pins A and B must always remain in the vertical slot of yoke C, which moves to the right at a constant speed of 6 cm/s. Furthermore, the pins cannot leave the elliptic slot. What is the speed at which the pins approach each other when the yoke slot is at x = 50 cm? What is the rate of change of speed toward each other when the yoke slot is again at x = 50 cm?
100 cm 60 cm
x
6 cm/s
yoke
vx =6 cm/s (constant), pins cannot leave the elliptic slot. What is the speed at which the pins approach each other when the yoke slot is at x = 50 cm? What is the rate of change of speed toward each other when the yoke slot is again at x
= 50 cm?
100 cm 60 cm
x
6 cm/s
yoke
10. A projectile is launched with speed v0 from
point A. Determine the launch angle q which results
in the maximum range R up the incline of angle a
(where 0≤ a ≤ 90°). Evaluate your results for a = 0,
30° and 45°.
launch angle q for maximum R, evaluate results for a = 0, 30° and 45°
x
y
0cos2
costancossin0
cos
cos
2
1tancossin
cos
cos
2
1
cos
cossinsin)2(
cos
cos)1(
)2(2
1sinsin
2
1
)1(coscos
220
2
220
22
2
000
0
20
200
000
q
aqaa
q
aqaa
q
a
q
aqa
q
a
qa
qa
v
gRR
v
RgRR
v
Rg
v
RvRinputting
v
Rtfrom
gttvRBatgttvyy
tvRBattvxx
y
x
aqa
q
qaaa
q
aa
tantancos
cos2
0tancossincos
cos2
22
coscos
2
22
g
vR
Rg
v
o
o
a
q2
220
cos
cos2
g
v
acosR
asinR
launch angle q for maximum R, evaluate results for a = 0, 30° and 45°
x
y
0
sco
1cos
sco2tantan
cos
sincos40
2
22
0
2
0
qa
qaq
a
q g
v
g
v
ddR
aq
aq
aaq
qaq
qaqqaqq
qqq
qaqq
qaqqa
tan
12tan0
tan
12tan
tan
101tan2tan,
2cos
1
tan
1arctan2
02costan2sin01sin2tansincos2
sin212cos01cos
sintansincos2
01tantansincos2cos
2
2cos
2
2sin
2
2
0
g
v
q
q2
2
cos
1sec
tan
d
d
aqa
qtantan
cos
cos2 22
g
vR o
q2cos1
launch angle q for maximum R, evaluate results for a = 0, 30° and 45°
x
y
2
90 aq
aaa
q
9090180
tan
1arctan1802
5.6745
6030
450
qa
qa
qa
aq
tan
12tan
since it is (-) it should be less than 180°
aa
aa
aaaa
90tan
1arctan
)90tan(tancottan
)90tan(cotcottan
1
1
arctan
1
and
All sine
tangent cosine
