82 CHAPTER 4 MOTION IN TWO AND THREE DIMENSIONS
Figure 4-23Question 5.(a)(b)(c)Projectile Motion Projectile motionis the motion of a particle
that is launched with an initial velocity. During its flight, the par-
ticle’s horizontal acceleration is zero and its vertical acceleration is
the free-fall acceleration g.(Upward is taken to be a positive di-
rection.) If is expressed as a magnitude (the speed v 0 ) and an an-
gleu 0 (measured from the horizontal), the particle’s equations of
motion along the horizontal xaxis and vertical yaxis are:v 0:v 0Questions
1 Figure 4-21 shows the path taken by
a skunk foraging for trash food, from
initial point i. The skunk took the same
timeTto go from each labeled point to
the next along its path. Rank points a,b,
andcaccording to the magnitude of the
average velocity of the skunk to reach
them from initial point i, greatest first.
2 Figure 4-22 shows the initial posi-
tioniand the final position fof a parti-
cle. What are the (a) initial position
vector :riand (b) final position vector , both in unit-vector nota-r:fFigure 4-21
Question 1.aib cFigure 4-22Question 2.zxify4 m4 m1 m2 m3 m3 m3 m 5 mtwice as long as at 45º. Does that result mean that the air density at
high altitudes increases with altitude or decreases?
4 You are to launch a rocket, from just above the ground, with
one of the following initial velocity vectors: (1) ,
(2) , (3) , (4). In
your coordinate system,xruns along level ground and yincreases
upward. (a) Rank the vectors according to the launch speed of the
projectile, greatest first. (b) Rank the vectors according to the time
of flight of the projectile, greatest first.
5 Figure 4-23 shows three situations in which identical projectiles
are launched (at the same level) at identical initial speeds and an-
gles. The projectiles do not land on the same terrain, however.
Rank the situations according to the final speeds of the projectiles
just before they land, greatest first.:v 0 20iˆ70jˆ :v 0 20iˆ70jˆ :v 0 20iˆ70jˆ:v 0 20iˆ70jˆUniform Circular Motion If a particle travels along a circle or
circular arc of radius rat constant speed v, it is said to be in uniform
circular motionand has an acceleration of constant magnitude(4-34)The direction of is toward the center of the circle or circular arc,
and is said to be centripetal.The time for the particle to complete
a circle is. (4-35)
Tis called the period of revolution,or simply the period,of the
motion.Relative Motion When two frames of reference AandBare
moving relative to each other at constant velocity, the velocity of a par-
ticlePas measured by an observer in frame Ausually differs from that
measured from frame B. The two measured velocities are related by
(4-44)where is the velocity of Bwith respect to A. Both observers
measure the same acceleration for the particle:
:aPA:aPB. (4-45):vBA:vPAv:PB:vBA,T2
r
va::aa
v^2
r
.:axx 0 (v 0 cosu 0 )t, (4-21)
, (4-22)
vyv 0 sinu 0 gt, (4-23). (4-24)
The trajectory(path) of a particle in projectile motion is parabolic
and is given by
, (4-25)
ifx 0 andy 0 of Eqs. 4-21 to 4-24 are zero. The particle’s horizontal
rangeR, which is the horizontal distance from the launch point to
the point at which the particle returns to the launch height, isR (4-26)v^20
g
sin 2 0.y(tan 0 )xgx^2
2(v 0 cos 0 )^2v^2 y(v 0 sin 0 )^2 2 g(yy 0 )yy 0 (v 0 sin 0 )t^12 gt^23 When Paris was shelled from 100 km away with the WWI
long-range artillery piece “Big Bertha,” the shells were fired at an
angle greater than 45º to give them a greater range, possibly even
6 The only good use of a fruitcake
is in catapult practice. Curve 1 in
Fig. 4-24 gives the height yof a cata-
pulted fruitcake versus the angle u
between its velocity vector and its
acceleration vector during flight. (a)
Which of the lettered points on that
curve corresponds to the landing of
the fruitcake on the ground? (b)
Curve 2 is a similar plot for the sameyθ
AB21Figure 4-24Question 6.tion? (c) What is the xcomponent of displacement :r?