MAT 265 – Brewer
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.
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:
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.
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 :
The curve created by the movement
of the ballerina during a jeté covered an area of
approximately 1.416430 ft2.