We will show later in the text that if a projectile is fired from ground level with an initial speed of υ 0 meters per second at an angle α with the horizontal, and if air resistance is neglected, then its position after t seconds, relative to the coordinate system in the accompanying figure is x = υ 0 cos α t , y = υ 0 sin α t − 1 2 g t 2 where g ≈ 9.8 m / s 2 . (a) By eliminating the parameter, show that the trajectory lies on the graph of a quadratic polynomial. (b) Use a graphing utility to sketch the trajectory if α = 30 ° and υ o = 1000 m / s . (c) Using the trajectory in part (b), how high does the shell rise? (d) Using the trajectory in part (b), how far does the shell travel horizontally?
We will show later in the text that if a projectile is fired from ground level with an initial speed of υ 0 meters per second at an angle α with the horizontal, and if air resistance is neglected, then its position after t seconds, relative to the coordinate system in the accompanying figure is x = υ 0 cos α t , y = υ 0 sin α t − 1 2 g t 2 where g ≈ 9.8 m / s 2 . (a) By eliminating the parameter, show that the trajectory lies on the graph of a quadratic polynomial. (b) Use a graphing utility to sketch the trajectory if α = 30 ° and υ o = 1000 m / s . (c) Using the trajectory in part (b), how high does the shell rise? (d) Using the trajectory in part (b), how far does the shell travel horizontally?
We will show later in the text that if a projectile is fired from ground level with an initial speed of
υ
0
meters per second at an angle
α
with the horizontal, and if air resistance is neglected, then its position after t seconds, relative to the coordinate system in the accompanying figure is
x
=
υ
0
cos
α
t
,
y
=
υ
0
sin
α
t
−
1
2
g
t
2
where
g
≈
9.8
m
/
s
2
.
(a) By eliminating the parameter, show that the trajectory lies on the graph of a quadratic polynomial.
(b) Use a graphing utility to sketch the trajectory if
α
=
30
°
and
υ
o
=
1000
m
/
s
.
(c) Using the trajectory in part (b), how high does the shell rise?
(d) Using the trajectory in part (b), how far does the shell travel horizontally?
System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.
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