PROBLEM 11.149
A child throws a ball from point A with an initial
velocity v0 at an angle of 3 with the horizontal.
Knowing that the ball hits a wall at point B,
determine (a) the magnitude of the initial velocity,
(b) the minimum radius of curvature of the
trajectory.
PROBLEM 11.150
A projectile is fired from Point A
with an initial velocity
0
.
v
(a)
Show that the radius of curvature
of the trajectory of the projectile
reaches its minimum value at the
highest Point B of the trajectory.
(b) Denoting by
the angle
formed by the trajectory and the
horizontal at a given Point C, show
that the radius of curvature of the
trajectory at C is 3
min
/cos .
3
cos
C
PROBLEM 11.151*
Determine the radius of curvature of the path described by the particle of Problem 11.95 when 0.t
PROBLEM 11.95 The three-dimensional motion of a particle is defined by the position vector
r (Rt cos
nt)i ctj (Rt sin
nt)k. Determine the magnitudes of the velocity and acceleration of the
particle. (The space curve described by the particle is a conic helix.)
SOLUTION
R
R
R
R
dRtttcRttt
r
2n
R
PROBLEM 11.152*
Determine the radius of curvature of the path described by the particle
of Problem 11.96 when 0,t
A
3, and 1.B
PROBLEM 11.96
The three-dimensional motion of a particle is defined by the position
vector 2
(cos)( )(sin),
1
A
tt A Btt
t
ri
j
k where r and t are
expressed in feet and seconds, respectively. Show that the curve
described by the particle lies on the hyperboloid (y/A)2 (x/A)2
(z/B)2 1. For 3A
and 1,B
determine (a) the magnitudes of the
velocity and acceleration when 0,t
(b) the smallest nonzero value
of t for which the position vector and the velocity are perpendicular to
each other.
42 242 3 1/2
2[ 19 1 8(cos sin ) 8( )sin 2 ]
dt tt tttttt
PROBLEM 11.152* (Continued)
2
22
PROBLEM 11.153
A satellite will travel indefinitely in a circular orbit around a planet if the normal component of the
acceleration of the satellite is equal to 2
(/)
g
Rr , where g is the acceleration of gravity at the surface of the
planet, R is the radius of the planet, and r is the distance from the center of the planet to the satellite. Knowing
that the diameter of the sun is 1.39 Gm and that the acceleration of gravity at its surface is 2
274 m/s ,
determine the radius of the orbit of the indicated planet around the sun assuming that the orbit is circular.
Earth: mean orbit
( ) 107u Mm/h.
(29.72 10 )
149.8 Gmr
PROBLEM 11.154
A satellite will travel indefinitely in a circular orbit around a planet if the normal component of the
acceleration of the satellite is equal to 2
(/),
g
Rr where g is the acceleration of gravity at the surface of the
planet, R is the radius of the planet, and r is the distance from the center of the planet to the satellite. Knowing
that the diameter of the sun is 1.39 Gm and that the acceleration of gravity at its surface is 2
274 m/s ,
determine the radius of the orbit of the indicated planet around the sun assuming that the orbit is circular.
Saturn: mean orbit
( ) 34.7u Mm/h.
SOLUTION
32
(9.639 10 )
1425 Gmr
PROBLEM 11.155
Determine the speed of a satellite relative to the indicated planet if the satellite is to travel indefinitely in a
circular orbit 100 mi above the surface of the planet. (See information given in Problems 11.153–11.154).
Venus: 2
29.20 ft/s ,g 3761R mi.
SOLUTION
2
g
R
g
PROBLEM 11.156
Determine the speed of a satellite relative to the indicated planet if the satellite is to travel indefinitely in a
circular orbit 100 mi above the surface of the planet. (See information given in Problems 11.153–11.154).
Mars: 2
12.17 ft/s ,g 2102R mi.
SOLUTION
g
2
g
R
PROBLEM 11.157
Determine the speed of a satellite relative to the indicated planet if the satellite is to travel indefinitely in a
circular orbit 100 mi above the surface of the planet. (See information given in Problems 11.153–11.154).
Jupiter: 2
75.35 ft/s ,g 44,432R mi.
SOLUTION
2
g
R
g
PROBLEM 11.158
A satellite will travel indefinitely in a circular orbit around the earth if the normal component of its
acceleration is equal to
2
/
g
Rr, where 2
9.81 m/sg, R = radius of the earth = 6370 km, and r = distance
from the center of the earth to the satellite. Assuming that the orbit of the moon is a circle of radius
3
384 10 km, determine the speed of the moon relative to the earth.
PROBLEM 11.159
Knowing that the radius of the earth is 6370 km, determine the time of one orbit of the Hubble Space
Telescope, knowing that the telescope travels in a circular orbit 590 km above the surface of the earth.
(See information given in Problems 11.153–11.155.)
orbit
PROBLEM 11.160
Satellites A and B are traveling in the same plane in circular orbits around the
earth at altitudes of 120 and 200 mi, respectively. If at 0t the satellites are
aligned as shown and knowing that the radius of the earth is R3960 mi,
determine when the satellites will next be radially aligned. (See information
given in Problems 11.153–11.155.)
RR
gg
PROBLEM 11.160 (Continued)
C
Alternative solution
3/2 3/2
3/2 3/2
11
()()
1/2
21 mi
5280 ft
11
(3960 120) mi (3960 200) mi
2
(3960 mi) 32.2 ft/s
1
1 h
3600 s
AB
C
Rh Rh
Rg
or 51.2 h
C
T
PROBLEM 11.161
The oscillation of rod OA about O is defined by the relation 3/ sin t
where
and t are expressed in radians and seconds, respectively. Collar B slides
along the rod so that its distance from O is 2
6(1 )
t
re
where r and t
are expressed in inches and seconds, respectively. When t 1 s, determine (a) the
velocity of the collar, (b) the acceleration of the collar, (c) the acceleration of the
collar relative to the rod.
SOLUTION
Calculate the derivatives with respect to time.
2
2
3
6 6 in. sin rad
t
t
re t
/( 3.25 in/s )
BOA r
PROBLEM 11.162
The path of a particle P is a limaçon. The motion of the particle is
defined by the relations
2cosrb t
and ,t
where t and
are expressed in seconds and radians, respectively. Determine
(a) the velocity and the acceleration of the particle when 2t s,
(b) the value of
for which the magnitude of the velocity is
maximum.
, hencet
PROBLEM 11.163
During a parasailing ride, the boat is traveling at
a constant 30 km/hr with a 200 m long tow line.
At the instant shown, the angle between the line
and the water is 30º and is increasing at a
constant rate of 2º/s. Determine the velocity and
acceleration of the parasailer at this instant.
SOLUTION
P
PROBLEM 11.164
Some parasailing systems use a winch to pull the rider back to the boat. During the interval when
is
between 20º and 40º, (where t = 0 at
= 20º) the angle increases at the constant rate of 2 º/s. During this time,
the length of the rope is defined by the relationship 600
/, where r and t are expressed in ft and s,
respectively. Knowing that the boat is travelling at a constant rate of 15 knots (where 1 knot = 1.15 mi/h),
(a) plot the magnitude of the velocity of the parasailer as a function of time (b) determine the magnitude of
the acceleration of the parasailer when t = 5 s.
SOLUTION
1.15mph 5280 /
ft mi
aee
PPx Py
Prr
aa
PROBLEM 11.164 (Continued)
P
PROBLEM 11.165
As rod OA rotates, pin P moves along the parabola BCD. Knowing that the
equation of this parabola is 2/(1 cos )rb
and that ,kt
determine the
velocity and acceleration of P when (a) 0,
(b) 90 .
SOLUTION
2
1cos
2sin 0
b
rkt
kt
bk kt
r