Unformatted text preview: Section 3.5 Projectile Motion 39. 40. 41. 42. 43. You toss an apple horizontally at 8.7 m/s from a height of 2.6 m.
Simultaneously, you drop a peach from the same height. How
long does each take to reach the ground? A carpenter tosses a shingle off an 8.8—m—high roof, giving it an
initially horizontal velocity of 11 m/s. (a) How long does it take
the shingle to reach the ground? (b) How far does it move hori—
zontally in this time? An arrow ﬁred horizonmlly at 41 mls travels 23 m horizontally
before it hits the ground. From what height was it ﬁred? Ink droplets in an ink-jet printer are ejected horizontally at 12 mfs
and travel a horizontal distance of 1.0 mm to the paper. How far
do they fall in this interval? Protons in a particle accelerator drop 1.2 ,um over the 1.7—krn
length of the accelerator. What is their approximate average
speed? If you can hit a golf ball 180 m on Earth, how far can you hit it on
the Moon? (Your answer is an underestimate because the distance
on Earth is restricted by air resishance as well as by a larger 3.) Se’étion 3. 6 Uniform Circular Motion t45' 46. 47. How fast would a car have to round a turn with a radius of 75 m
for its acceleration to be numerically equal to that of gravity?
Estimate the acceleration of the Moon, which completes a nearly
circular orbit of 385,000 km radius in 27 days. Global Positioning System satellites are at altitudes of approxie
mately 20,000 km, where the acceleration of gravity is only 5.8%
of its surface value. To the nearest hour, what’s the orbital period
of the GPS satellites? Problems 48. 49. 50 W1 52. \53‘ 54. Two vectors A and B” have the same magnitude A and are at right
angles. Find the magnitude of the vectors (a) A + 23 and
(b) 3A 15‘ Vector A has magnitude 1.0 m and points at 35° clockwise from
the x axis. Vector B has magnitude 1 S In What angle Should B
make with the x axis so that A + B is purely vertical? Let A —152—40] and B— —3l]+18k Find a vector C such that
A "1* B + C: 0. A biologist studying the motion of bacteria notes a bacterium at
position r1: 2_. 21+ 3. 7j-1 .216 um(1,u.m=106m).After6.2 s
the bacterium is at 13— 4 6i + 1_ ._9_k pm. What is its average ve
locity? Express the ahSwer in unit vectdr notation and calculate
the magnitude An object’s position as a function of time is given by
F = 1215+ (15t— 5.013)} m, where t is time in seconds. (a) What
is the object’s position at 1‘: 2.0 s? (b) What is its average veloc-
ity in the interval from t = 0 to I: 2.0 s? (c) What is its instanta-
neous velocity at I = 2.0 s? Attempting to stop on a slippery road, a car moving at 80 kmfh
skids across the road at a 30° angle to its inititd motion, coming to
a stop in 3.9 3. Determine the average acceleration in mst, using
a coordinate system with the x axis in the direction of the car’s
original motion and the y axis toward the side of the road to
which the car skids. An object undergoes acceleration of 233-1- 36} mls2 over a 10-5
interval. At the end of this time, its velocity is 33i+15fmfs.
(a) What was its velocity at the beginning of the 10-3 interval?
(b) By how much did its speed change? (c) By how much did its
direction change? (d) Show that the speed change is not given by
the magnitude of the acceleration multiplied by the time. Why not? 55. 56. 57. 58. 59. 60. 61. 62. 63. Exercises and Problems 47 The sweep-second hand of a clock is 3.1 cm long. What are the
magnitude of (a) the average velocity and (b) the average accelerw
ation of the hand’s tip over a 50—5 interval? (c) What is the angle
between the average velocity and acceleration vectors? A ferryboat sails between two towns directly opposite each other
on a river. If the boat sails at 15 km/h relative to the water, and if
the current ﬂows at 6.3 krnlh at what angle should the boat head?
The sum, A + B, of two vectors is perpendicular to the difference
A B How do the magnitudes of the two vectors compare?
Write an expression for a unit vector that lies at 45° between the
positive .76 and y axes. An object is moving initially in the x direction at 4.5 m/s, when
an acceleration is applied in the y direction for a period of 18 5. If
the object moves equal distances in the x and y directions during
this time, what is the magnitude of its acceleration? A particle leaves the origin with initial velocity in =
113+ 14} m/s. It undergoes a constant acceleration given by
a = 71.23+0.26j this? (a) When does the particle cross the y
axis? (b) What is its y coordinate at the time? (c) How fast is it
moving, and in what direction, at that time? A kid ﬁres water horizontally from a squirt gun held 1.6 m above
the ground. It hits another kid 2.1 m away square in the back, at
a point 0.93 m above the ground. What was the initial speed of
the water? In a chase scene, a movie Stuntman is supposed to run right off
the ﬂat roof of one city building and land on another roof 1.9 m
lower. If the gap between the buildings is 4.5 In wide, how fast
must he run? Standing on the ground 3.0 m from the wall of a building, you
want to throw a package from your 1.5—m shoulder level to some
one in a second-ﬂoor window 4.2 In above the ground. At what
speed and angle should you throw the package so it just barely . reaches the window?
6\ Derive a general formula for the horizontal distance covered by a 65. 66. 67. 68. 69. 70. proy ectile launched horizontally at speed v0 from height h. Compare the travel times for the projectiles launched at 30° and 60"
in Fig. 3.19, both of which have the same starting and ending points.
You toss a chocolate bar to your hiking companion located 8.6 m
up a 39U slope, as shown in Fig. 3.23. Determine the initial veloc~
ity vector so that the chocolate bar will reach your friend moving
horizontally. Elﬁllllﬂli Problem 66 Prove that a projectile launched on level ground reaches its max-
imum height midway along its trajectory. A projectile launched at an angle 0 to the horizontal reaches a
maximum height it. Show that its horizontal range is 4h/tan 0. A motorcyclist driving in a 60 lcmlh zone hits a stopped car on a
level road. The cyclist is thrown from his bike and lands 39 In
down the road. Was the cyclist speeding? To answer, ﬁnd the min-
imum Speed he could have been going just before the accident.
Show that, for a given initial speed, the horizontal range of a pro—
jectile is the same for launch angles 45" + a: and 45° 7 a, where a
is between 0° and 45°. ...
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- Spring '07