Section 1.1: Using Hang Time to Estimate Jump Height

Measuring a jump's height can be tricky without a nearby reference point. In gymnastics, during a floor routine, there’s often nothing adjacent to the athlete as they perform. However, we can approximate the jump height by analyzing the "hang time," which refers to the duration from when the gymnast leaves the ground until they return. Once airborne, the only force acting on them is gravity, making this a classic case of projectile motion.

In projectile motion, the gymnast experiences a vertical acceleration of –9.8 meters per second squared (a constant known as g). Using this acceleration, we can apply a kinematic equation:

In this equation, y represents the final position, and y0 is the starting position. As Simone begins and ends on the floor, both positions are equal (let’s assume y = y0 = 0 meters). While this doesn't help us find the maximum height directly, we can employ a clever approach by analyzing her jump mechanics.

Starting with a vertical velocity v0, she ascends until her velocity reaches zero at the peak height. As she descends, her velocity matches her initial upward velocity in magnitude but is directed downward. By dividing the total time (t) into two equal halves, we can analyze the ascent separately.

Subsection 1.1.1: Kinematic Equations and Height Calculation

At her maximum height (h), where the initial velocity is zero, we can adapt our kinematic equation to solve for h:

This equation helps us understand the relationship between hang time and maximum height.

Section 1.2: Video Analysis for Accurate Measurements

To gain a more precise understanding of Simone’s jump, we can analyze her performance using video footage.

In this analysis, I used Tracker Video Analysis (a free tool) to track Simone's motion throughout the triple double. By marking her position in each video frame, I can generate position-time data. Although the camera's movements complicate measurements, we can still extract the hang time.

In my observations, the time from when Simone leaves the floor to when she lands is approximately 1.235 seconds, which corresponds to an estimated jump height of 1.87 meters (about 6.2 feet). This figure, while impressive, is significantly less than the claimed 12 feet.

Another Video Analysis

To further explore this, we can analyze another video of Simone’s tumbling pass, which offers a better perspective for video analysis.

In this clip, the camera's stable position allows for clearer measurements. By tracking her motion with a coordinate system, I recorded a hang time of about 1.133 seconds. This indicates her center of mass reaches a peak of 1.62 meters.

After measuring from the ground to her head, I found a height of approximately 3.037 meters, or 9.96 feet. While this is still shy of the 12-foot mark, it illustrates the remarkable heights she can achieve.

Homework

As an exercise, consider the following questions:

1. In her triple double, Simone rotates around her hips twice and her spine three times. Estimate her angular velocity based on her flight time.
2. If Simone aimed for a jump height of 13 feet, what would her corresponding hang time need to be?
3. Calculate her initial vertical velocity during the "12 feet" jump.
4. Estimate the time it takes for her to land, and calculate the average force experienced during the impact, using her mass for these calculations.