The Fascinating World of Penguin Jumps

Penguins leaping from towering ice cliffs is a captivating sight, as showcased in an exciting National Geographic trailer. This remarkable footage offers a unique opportunity to delve into the physics behind such jumps.

The premise is that these penguins must leap off a substantial ice ledge to reach the water below. A striking aerial shot captures this action, leading us to an intriguing physics question: How Tall Is the Cliff? While the video claims it's a 50-foot drop, I aim to analyze it independently.

To begin, I will utilize Tracker Video Analysis. For those unfamiliar, this technique involves tracking the position of an object frame by frame in a video, resulting in position-time data. For a falling object, such as a penguin, there are three key variables to consider:

1. Frame Rate: This indicates the time interval between frames.
2. Distance Scale: An object of known size is required to establish a pixel-to-distance ratio.
3. Vertical Acceleration: We can assume that the penguin experiences an acceleration of -9.8 meters per second², the standard for objects on Earth.

Assuming the penguin's jump is depicted in real time (which isn't the case in slow-motion shots), I can correlate the time and acceleration to calculate the distance scale, which ultimately leads to determining the height of the ice cliff.

Most of the jumps feature some minor camera movement; however, I managed to find a jump with minimal motion interference.

For the distance scale, I have opted to use a measurement where one unit corresponds to the cliff's height, although I need to convert this to meters. Below is the graph depicting the vertical position of the penguin as it descends towards the water.

The plotted data appears to follow a parabolic trajectory, indicative of constant acceleration behavior, which is promising. This suggests that the position versus time relationship can be expressed by the following function.

This leads us to determine that the coefficient associated with the t² term in the fitted curve equals -1/2*g. Consequently, I derive:

Thus, one unit of distance corresponds to approximately 9.28 meters. Now, we can calculate how far the penguin actually drops. Analyzing the data, the penguin began at y = 1.119 units and ended at 0.0 units. Converting this measurement into meters indicates a drop of 10.38 meters. To align with the video's claim, we should convert this to feet: 10.38 meters equals about 34.1 feet.

It seems National Geographic may have exaggerated! Although they state it’s a 50-foot drop, I suspect they rounded up or relied on the drone's altitude for estimation. Alternatively, the video could be edited for non-real-time playback.

Regardless, I will stand by my calculation: the jump measures 10.38 meters.

Further Exploration

There are additional questions raised by this analysis that could serve as intriguing homework assignments:

1. How fast was the penguin traveling upon impact with the water?
2. What is the terminal velocity of an emperor penguin? How does air resistance factor into this for both emperor and standard-sized penguins?
3. If the penguin continues downward after hitting the water's surface, how deep does it go? Estimate its acceleration as a g-force.
4. If the penguin executes a sharp horizontal turn underwater, and the radius of that turn is 1 meter, what would be the estimated acceleration?
5. Finally, if the penguin dashed at maximum speed (a homework question in itself, as I’m unsure of their running speed), how far could it land from the base of the ice cliff?