Unformatted text preview: −2v0 sin θ −2v sin θ
0
x = (v0 cos θ ) ay and ay Using the fact that 2sin θ cos θ = sin 2θ , we have
x=− 2
2v0 cos θ sin θ ay =− 2
v0 sin 2θ ay Thus, we find that sin 2θ = −
and x ay
2
v0 =− (91.4 m) (–9.80 m/s 2 )
= 4.91× 10−3
2
(427 m/s) 2θ = 0.281° or 2θ = 180.000° − 0.281° = 179.719° Therefore, θ = 0.141° and 89.860°
_____________________________________________________________________________________________ Chapter 3 Problems 131 48. REASONING The angle θ is the angle that the balloon’s initial velocity v0 makes with the
horizontal, and can be found from the horizontal and vertical components of the initial
velocity:
v0 y v0 y (1)
tan θ =
or
θ = tan −1 v v0 x 0x (Note: because the balloon follows a curved trajectory, θ is not related in this fashion to the
horizontal and vertical components x and y of the balloon’s displacement.) We will use
x = v0 x t + 1 a x t 2 (Equation 3.5a) and y = v0 yt + 1 ayt 2 (Equation 3.5b) to find expressions
2
2 for the horizontal and vertical components (v0x and v0y) of the balloon’s initial velocity, and
then combine those results with Equation (1) to find θ. For the initial horizontal velocity
component, with ax = 0 m/s2 since air resistance is being ignored, Equation 3.5a gives ( ) x = v0 x t + 1 0 m/s 2 t 2 = v0 x t
2 v0 x = or x
t (2) For the initial vertical velocity component, we obtain from Equation 3.5b that y = v0 y t + 1 a t2
2y or v0 y = y − 1 a yt 2
2
t (3) In part b, the initial speed v0 of the second balloon can be found by first noting that it is
related to the x component v0x of the balloon’s initial velocity by v0 x = v0 cos θ . Since there
is no acceleration in the x direction (ax = 0 m/s2), v0x is equal to the x component of the
balloon’s displacement (x = +35.0 m) divided by the time t that the second balloon is in the
air, or v0x = x/t. Thus, the relation v0 x = v0 cos θ can be written as x
= v0 cosθ
t (4) In Equation (4), x = +35.0 m and θ is known from the result of part (a). The time of flight t
is related to the y component of the balloon’s displacement by y = v0 yt + 1 ayt 2 (Equation
2
3.5b). Since v0 y = v0 sin θ , Equation 3.5b can be expressed as y = v0 yt + 1 a y t 2 = ( v0 sin θ ) t + 1 a yt 2
2
2 (5) Equations (4) and (5) provide everything necessary to find the initial speed v0 of the water
balloon for the second launch. 132 KINEMATICS IN TWO DIMENSIONS SOLUTION
a. Substituting Equations (2) for v0x and (3) for v0y into Equation (1) gives an expression for
the balloon’s initial direction θ : y − 1 a yt 2 2 y − 1 a yt 2 v0 y t
2
−1
−1 = tan −1 θ = tan = tan x v0 x x t (5) The balloon leaves the roof of Jackson, 15.0 m above the ground, and hits halfway up
Walton, a point 1 (22.0 m) = 11.0 m above the ground. The balloon’s height at impact is
2
therefore 15.0 m − 11.0 m = 4.0 m below its launch height. Taking up as the positive
direction, the vertical displacement of the balloon is therefore y = −4.0 m, and its horizontal
displacement is x = +35.0 m, the distance between the buildings. Therefore, the angle at
which the first balloon is launched is ( ) −4.0 m − 1 −9.80 m/s 2 ( 2.0 s )2 2 = 24o
θ = tan −1 35.0 m b. The angle θ is that found in part (a), and the vertical displacement is the difference in
heights between the Walton & Jackson dorms: y = 22.0 m − 15.0 m = +7.0 m. Solving
Equation (4) for the elapsed time t and substituting the result into Equation (5) gives y = ( v0 sin θ ) t + 1 a t2
2y x 1 x = ( v0 sin θ ) + 2 ay v0 cos θ v0 cos θ 2 Solving this equation for the initial speed v0 of the second balloon gives
x
v0 =
cos θ ( ) 1 −9.80 m/s2
35.0 m
2
=
= 29 m/s
sin θ
cos 24° 7.0 m − 35.0 m sin 24°
y−x
(
)
cosθ
cos 24°
1a
2y 49. SSM REASONING Since the horizontal motion is not accelerated, we know that the x
component of the velocity remains constant at 340 m/s. Thus, we can use Equation 3.5a
(with ax = 0 m/s2) to determine the time that the bullet spends in the building before it is
embedded in the wall. Since we know the vertical displacement of the bullet after it enters
the building, we can use the flight time in the building and Equation 3.5b to find the y Chapter 3 Problems 133 component of the velocity of the bullet as it enters the window. Then, Equation 3.6b can be
used (with v0 y = 0 m/s) to determine the vertical displacement y of the bullet as it passes
between the buildings. We can determine the distance H by adding the magnitude of y to the
vertical distance of 0.50 m within the building.
Once we know the vertical displacement of the bullet as it passes between the buildings, we
can determine the time t1 required for the bullet to reach the window using Equation 3.4b.
Since the motion in the x direction is not accelerated, the distance D can then be found from
D = v0 xt1 .
SOLUTION Assuming that the direction to the right is positive, we find that the time that
the bullet spends in the building is (according to Equation 3.5a) t= x
v0 x = 6.9 m
= 0.0203 s
340 m/s The vertical displacement of...
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 Spring '13
 CHASTAIN
 Physics, Pythagorean Theorem, Force, The Lottery, Right triangle, triangle

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