Kristell Millán
MAT 265 – Brewer
TTH/
Calculus Applications in the Realm of Dance
The most revered aspects of a ballet are often the beautiful jetés executed by the ballerinas. These leaps through the air result in a beautiful picture with the ballerina’s legs completely extended in front of and behind her, a picture which often leaves people in awe of the height and grace of this common dance leap. However, there is more to this leap, and other dance moves, than just poise. There are also patterns that can be explained by the application of Calculus.
Part I
A
ballerina leaping through the air in a jeté is
assumed to be a projectile. Visualizing the ballerina’s
movement through the air as a function that can be plotted on a graph, we see
that the jeté constitutes movement in two directions,
the x-direction and the y-direction. The movement in the x-direction can be
described by the function:
where the
velocity in the x-direction 
.
The movement in the y-direction can
be described by the function:
where the
velocity in the y-direction 
Assumptions and estimated
measurements:
·
Average height of a ballerina is
5’7” with hip height 
at ≈ 3.5 ft. in height.
·
Initial angle created by the
ballerina’s legs at the beginning of the jeté 
is assumed to be 40°. This measurement is
highly variable as a result of style, flexibility, and individual height. Other
reasonable possibilities for this measurement would be between 30° and 55°.
·
Force of gravity on dancer
.
·
To calculate 
, the
starting speed, we must first make a logical estimated measurement of 
. Dancers
usually go into a jeté from a short chassé run. I did this myself several times, measuring the
distance traveled as well as the time it took. I found that the average was
about 7 ft/s. With this estimate for we can then measure the value of using right triangle trigonometry:
Plugging in our the values we know, 
Given these constants:
Part II
Maximum Height: To find the maximum height achieved by a ballerina
during a jeté we set 
to find the critical numbers and then plug in
the resulting critical numbers into the original equation 
to find the maximum value.
To find critical numbers:
To check that this is where a
maximum occurs, we evaluate 
for 
and 
. is
positive and 
is negative so the function
must have been increasing from
(-∞ ,
.183553) and decreasing from (.183553 , ∞). This means that .183553 is
the 
value at which a maximum occurs.
Plugging back this critical number into the original equation, we calculate that the maximum height achieved by the ballerina in her jeté is:
The ballerina achieves maximum
height at 
seconds when the ballerina’s hip is 4.039068
feet high.
Slant of Ballerina’s Legs: Assuming
the ballerina’s legs to be in perfect splits, the slant of her legs can be
defined as the slope of the tangent of the tangent line 
.
By plugging in different values for
we can evaluate the slope of the ballerina’s
legs at any moment during the jeté.
At = 0:
This seems reasonable for the slope of the ballerina’s legs at start because at 45° the slope would be 1 but since she starts at 40° the slope should be slightly less than 1.
Also, at = .183553:
Because at = .183553 there was a maximum,
we would expect the slope to be zero. Because of rounding in previous steps, we
do not get zero exactly but the answer we do get is very close.
We can also calculate at what time the ballerina finishes the jeté
by setting 
because the ballerina’s legs should be at the
same slant as when she started but now with a negative slope.
Part III
Area Covered Under the Curve- The Integral: The motion of the
ballerina as she does a jeté through the air creates
a curve that is described by the function 
. By evaluating the integral,
we can accurately calculate the area covered under this curve.
We know the jeté
starts at 
but we need to find out at what time it ends.
To do this, we set 
:
.367106
The jeté is completed after .367106
seconds. Therefore, the integral:
The curve created by the movement
of the ballerina during a jeté covered an area of
approximately 1.416430 ft2.