Greenwood press

4
ANGLE
The path of a boat from its starting
position, S, to its ending position, E,
when it is sailing against a headwind.
The angle of a ball
approaching a wall will
equal the angle of the
ball leaving the wall,
assuming there is no 
spin on the ball.

---

yields
1
4.9
v
2
0
sin a cos a. Using trigonometric identities gives x =
v
2
0
sin(2α)
9.8
.
Suppose a golfer hits a tee shot, and that his or her club hits the ball at 
v
0
= 70
meters/second. The graph of horizontal distances x as a function of the angle 
α
shows that the angle that will give the golfer the best distance is 45° (
π/4 radi-
ans). Frogs know this angle: push-off angle for a frog hop has been measured to
be close to 45°.
When a golfer tees off or a football kicker aims for a long field goal, he or
she should strike the ball at a 45° angle in order to obtain maximum distance. A
baseball player, on the other hand, needs to alter this thinking slightly, because
he hits a ball about 1 meter off of the ground. This makes the horizontal-distance
equation more complicated: 
x = v
0
cos α


v
0
sin α
√
v
2
0
sin
2
α−19.6(y−1)
9.8

.
When the ball hits the ground (y = 0), the graph of this function shows that a ball
reaching the bat at 85 miles per hour, or 38 meters per second, will attain a max-
imum horizontal distance when the ball leaves the bat at about a 44.8° angle,
very close to the angle if the ball were hit from the ground.
The refraction of light is dependent on the angle in which light enters the
object and the material it passes through. Snell’s law states that 
n
1
• sin Θ
1
=
n
2
• sin Θ
2
, where 
n is the index of fraction (the ratio of the speed of light in air
to the speed of light in that material) and 
Θ is the angle of incidence. As light
passes through an object, such as a glass of water, it will bend, giving it a dis-
torted view if you look through the glass. Higher values of 
n allow the light to
bend more, since 
Θ
2
decreases as 
n
2
increases.