1. The velocity of a particle moving in the x-y plane is given by m/s at time t=3.65 s. Its average acceleration during the next 0.02 s is m/s 2 . Determine the velocity of the particle at t=3.67 s and the angle q between the average-acceleration vector and the velocity vector at t=3.67 s. Solution: at t=3.65 s, (during 0.02 s) at t=3.67 s, j i 24 . 3 12 . 6 v j i 6 4 ? 6 4 24 . 3 12 . 6 v j i a j i v av j i v v j i v j i t v a t t av 36 . 3 2 . 6 02 . 0 24 . 3 12 . 6 6 4 67 . 3 67 . 3 v a q 0 2 2 2 2 85 27 85 50 96 44 6 4 36 3 2 6 6 4 36 3 2 6 . . . cos cos . . j i j . i . cos a v a v q q q q
20
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Solution: t - DEUkisi.deu.edu.tr/binnur.goren/Dynamics2016G/4P_Cartesian... · y 51. 96 cm v y 2. 078 cm/s Acceleration Second derivative of the trajectory equation with respect to
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1. The velocity of a particle moving in the x-y plane is given by m/s
at time t=3.65 s. Its average acceleration during the next 0.02 s is m/s2.
Determine the velocity of the particle at t=3.67 s and the angle q between the
average-acceleration vector and the velocity vector at t=3.67 s.
Solution:
at t=3.65 s, (during 0.02 s)
at t=3.67 s,
ji
24.312.6
v
ji
64
?
6424.312.6
v
jiajiv av
jivv
jivji
t
va
t
tav
36.32.6
02.0
24.312.664
67.3
67.3
v
a
q
0
2222
85278550
9644
64363266436326
..
.cos
cos..jij.i.
cosavav
qq
q
q
2. The position of a particle is defined by Ԧ𝑟 = 5𝑐𝑜𝑠2𝑡Ԧ𝑖 + 4𝑠𝑖𝑛2𝑡Ԧ𝑗, [m], where t is
in seconds and the arguments for the sine and cosine are given in radians.
Determine the magnitudes of the velocity and acceleration of the particle when
t=1 s. Also prove that the path of the particle is elliptical.
3. The y-coordinate of a particle in curvilinear motion is given by y = 4t33t, where y is in meters
and t is in seconds. Also, the particle has an acceleration in the x-direction given by ax = 12t m/s2.
If the velocity of the particle in the x-direction is 4 m/s when t = 0, calculate the magnitudes of
the velocity and acceleration of the particle when t = 1 s. Construct and in your solution.
Solution:
when t=1 s vy = 9 m/s , ay = 24 m/s2
ax=12t
when t=1 s vx = 10 m/s , ax = 12 m/s2
v
a
v
a
taytvytty yy 2431234 23
4664
12
22
04
tvtv
tdtdvdtadvdt
dva
xx
tv
xxxx
x
x
a
ya
xaxvq
yv
v
o
x
y
yx
.v
vtana
s/m.vvv
9841
451322
q
o
x
y
yx
.a
atana
s/m.aaa
4363
8326 222
4. 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.
Solution:
x direction y direction
3
2
3
20
22
2
40
4
33
0
2
0
2
2
000
tx
tx
dttdxtdt
dxv
tv
dttvdtadv
tdt
dva
tx
x
x
t
x
t
x
v
x
xx
x
32
233
22
02
2
212
1
28
1
3
22
2
1
3
2
22
1
2442
8
88
yx
yyx
yt
tyty
tdtdydt
dyv
s/madt
dvtvy
/
ty
y
yy
y
Equation of the path
calculate the magnitude of the velocity of the particle for the instant when its x-
coordinate reaches 18 m.
smvvv
smvtv
smvtv
stt
xmx
yx
yy
xx
/302418
/248
/182
3183
218
2222
2
3
5. The skateboard rider leaves the ramp at A with an initial velocity vA at a 30o angle. If
he strikes the ground at B, determine vA and the time of flight.
6. The boy at A attempts to throw a ball over the roof of a barn such that it is launched at
an angle qA=40o. Determine the minimum speed vA at which he must throw the ball so
that it reaches its maximum height at C. Also, find the distance d where the boy must
stand so that he can make the throw.
7. Projectile (1) is fired with a speed of v=60 m/s at an angle of 60o. Projectile (2) is then
fired with the same speed 0.5 s later. Determine the angle q of the second projectile so
that the two projectiles collide. At what position (x,y) will this happen?
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 40 mm/s. All measurements are in mm and s. Calculate the magnitudes of Ԧ𝑣 and
Ԧ𝑎 of pin P when x = 60 mm.
SOLUTION
Trajectory of the pin is160
2xy ??60 avmmxwhen
jaiaajvivv yxyx
jaiaajvivv yxyx
yaxayvxv yxyx ,,
0)(/40 xacstsmmxv xx
The first derivative of trajectory equation with respect to time
smm
xxxxy
xy /30
80
4060
80160
2
160
2
jiv
3040 Magnitude of velocity of the pin: smmv /503040 22
22
2 /2080
40
80
1
80smmxxxy
xx
dt
dy
dt
d
ja
20
Magnitude of acceleration of the pin:
2/20 smma
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
Cx
y
100 cm
60 cm
x
6 cm/s
yoke
C x
y
Equation of elliptical slot is 160100 2
2
2
2
yx
??50 avcmxwhen
jaiaajvivv yxyx
cmycmxfor 96.5150
Yoke C moves at a constant speed, so 0/6 xvacstscmxv xxx
Using the equation of trajectory 160100 2
2
2
2
yx
First derivative of the trajectorywith respect to time
0
60
96.51
100
6500
60
2
100
22222
yyyxx
scmyvy /078.2 Velocity of pin B jivB
078.26
Velocity
(For pin A )scmvcmy y /078.296.51
Acceleration
Second derivative of the trajectory equation with respect to time
0
60
96.51
60
078.2
100
6
06060100100
22
2
2
2
2222
y
yyyyxxxx
2/3325.0 scmy
ja
3325.0 (For pin B)
10. A long-range artillery rifle at A is aimed at an angle of 45o with the
horizontal, and its shell is just able to clear the mountain peak at the top of its
trajectory. Determine the magnitude u of the muzzle velocity, the height H of the
mountain above sea level, and the range R to the sea.