Greenwood press

ANGLE
3
30° North of West
60° West of North
Alternate methods of denoting direction when a boat is 3 miles from
shore and 6 miles from its destination.
The navigation angle, 
β, of a
boat based on its distance from
shore, s, and the distance from its
final destination, d, is equal to
arccos(
s
d
).

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Angles are useful for reflecting light rays or objects off of flat objects. The
angle by which an object, such as a ball, approaches a wall is equal to the angle
by which the object bounces off the wall. This is true because a ball’s reflection
off a wall will be the same distance away from the wall as if it had gone in a
straight line. In essence, reflections preserve congruence. By the transitive prop-
erty, the angle of the ball coming into the wall will equal the angle of the ball
leaving the wall, as shown below. In billiards or miniature golf, a player can use
this principle when aiming for a hole by simply aiming for the hole’s reflection.
Athletes who try to throw or hit balls certain distances, such as baseballs, bas-
ketballs, footballs, and golf balls, use angles strategically. If they want to hit a
ball short and high, they will use an angle close to 90°. In order to hit a low-fly-
ing line drive, they will use an angle close to 0°. The horizontal distance in
meters, x, of an object can be determined by the product of its initial velocity in
meters per second, 
v
0
, the time in seconds, t, that the ball is in the air, and the
cosine of the angle, 
α, it is released or hit. Since the earth’s gravitational force
pulls a ball towards the surface, the vertical distance in meters, y, also needs to
be considered in order to determine the ideal angle at which to release or hit a
ball. The two equations describing the path of the ball in both directions are rep-
resented as
x = v
0
t cos α
y = v
0
t sin α − 4.9t
2
.
The ball will be on the ground when y is
0. Solving the second equation for the 
time
t that will provide this value gives t = 0 or t =
1
4.9
v
0
sin α. The latter solu-
tion gives the time the ball will be in the air. Substituting in the equation for x