Pom-Poms and Parabolas: The Physics of Cheerleading

Smith Cheerleaders Hana Hernandez and Louisa Miller-Out demonstrating a toe touch. Photo by Sadye Prebus.

What do cheerleaders and Sir Isaac Newton have in common? A robust understanding of the laws of physics! Based on his observations of the behavior of objects on Earth, Newton wrote three laws of motion. Cheerleaders understand these laws intuitively and apply them in order to execute difficult stunts. Gary Felder, a professor of physics at Smith, helped break down two foundational cheer skills: the toe touch and the basket toss.

Newton’s first law, or the law of inertia, states that an object in motion remains in motion and an object at rest remains at rest unless acted upon by an external force (F). If multiple forces are acting on an object, the net force–the sum of all the forces–must have a non-zero magnitude to cause it to move. 

The second law defines force as mass times acceleration (F = ma). Mass (m) quantifies the amount of matter in an object, and acceleration is change in velocity over time. Gravity causes a constant acceleration of 9.8 meters per second squared (denoted g) toward the center of the Earth, so the force of gravity (F_g) on an object or individual at any given time is mg. 

Newton’s third law of motion states that every force enacted by one object on another has an equal and opposite reaction force from the other object. One key example of a reaction force is the normal force (F_N), which balances gravity when an object is in contact with the Earth’s surface. Because the forces upon it are equal in magnitude and opposite in direction, the net force is zero, and the object does not accelerate.

Free-body diagram of the forces acting on a cheerleader during a toe touch. Created by Louisa Miller-Out using https://app.diagrams.net/.

The toe touch is a fundamental cheer skill, requiring strength, flexibility, and strategic deployment of Newtonian physics. At the beginning of a toe touch, when a cheerleader is at rest, the major forces acting on them are the constant gravitational force toward the ground and the normal force of the ground pushing back. These forces are equal in magnitude and opposite in direction. 

Felder explained that when the cheerleader bends their knees in preparation for the jump, they actively exert force on the ground with their muscles. This increases the reaction force of the ground on them until it exceeds the gravitational force, causing the cheerleader to accelerate upward until their feet leave the ground. At this point, they enter free-fall, meaning the only force affecting them is gravity. 

An object in free-fall travels in an arc called a parabola. Professor Felder noted that it is specifically the center of mass that traces this shape. An object’s center of mass is the point at which its mass can be considered most concentrated for mathematical purposes.  

In a toe touch, the parabola begins and ends in the same spot–there is only vertical displacement, not horizontal. Acceleration from the initial net force is highest at the beginning of the jump and decreases to zero during free-fall due to the downward pull of gravity. 

At the apex of the jump, the cheerleader pulls their legs upward and outward and extends their arms to touch their toes. This raises the position of their center of mass so that it is in their chest rather than their lower abdomen. Felder explained that this creates the illusion of “hovering” in the air because the torso remains in the same position for most of the jump despite the fact that the center of mass is only at its peak for a moment.

Diagram showing the parabolic path of a cheerleader’s center of mass during a toe touch. Created by Louisa Miller-Out using https://app.diagrams.net/.

When the downward force of gravity overcomes the cheerleader’s residual upward acceleration, their net acceleration flips from positive to negative, and they return to the ground. They absorb the force of impact by bending their knees. “This is also how airbags work,” Felder noted, explaining that they, too, increase the time interval over which a change in velocity takes place.

Smith cheerleaders preparing to catch the flyer during a basket toss. Photo courtesy of Louisa Miller-Out.

A basket toss can also be explained by physics. The Smith Cheer team does baskets in groups of five, with a backspot, two bases, a flyer, and a front spot. The bases and backspot bend their knees to create a net upward force and dip several times with the flyer perched on their hands to build momentum. Then they stand and throw their arms upward until the flyer leaves their hands and enters free-fall.

The flyer can shift their center of mass so their torso stays at maximum height for longer. In a straight-ride basket, the flyer extends their arms above their head as they leave the bases’ hands. At the apex of their flight, they pull their extended arms down in front of them and bring their feet up slightly, which causes the body to rotate backward about 90° during its descent. In more complex baskets, the flyer uses this principle to flip and twist, tucking their knees or pulling in their arms to rotate faster. 

A flyer in free-fall is able to modulate the speed of their rotation due to the conservation of angular momentum, which depends on mass, velocity, and the radius of a rotating body (L = mvr). The flyer’s mass is constant, so if their body becomes more compact, decreasing the radius of rotation, their velocity must increase in order to maintain a constant angular momentum. 

Conversely, in order to stop their rotation at a specific point in their trajectory, the flyer can open up and increase their radius of rotation, decreasing their velocity. 

The bases play a crucial role in determining the flyer’s trajectory and rotation. If they throw the flyer’s feet while they are in front of the rest of their body, this introduces torque, the rotational analog of force. Torque can be expressed as the force acting on an object multiplied by the distance of the contact point from the object’s center of mass (τ = Fr).

According to Felder, torque can be demonstrated by pulling downward on a door handle. If the flyer is pivoting around their center of mass, their limbs can act as levers to which torque can be applied, giving them the initial rotational acceleration needed to execute flips and twists. 

Cheerleading and classical mechanics may seem mystifying at first glance, but when one considers them in relation to one another, everything starts to fall into place. 

By Louisa Miller-Out