Skiing

This activity allows students to analyze the speed of a skier during an Olympic descent. It develops the ability to decompose a velocity vector and to understand the challenges of kinematic analysis.

During an Olympic downhill skiing event, athletes reach speeds exceeding 140 km/h while navigating icy turns on a steep mountainside. Television broadcasts display real-time speed readings that seem impossibly precise. But how is the speed actually measured, and can we verify it ourselves? Kinematic analysis of video provides a powerful method. However, there is a subtle challenge: a camera captures motion in two dimensions, but the skier moves in three. The speed displayed on screen is the true 3D speed, while direct analysis of video gives only the two components visible to the camera: horizontal and vertical. To reconstruct the true speed from a side-view video, students must combine these perpendicular components using the Pythagorean theorem: V = √(Vx² + Vy²). This experiment teaches students to decompose velocity into components and recombine them, a fundamental skill in physics and engineering.

Learning objectives:

The student uses the FizziQ Kinematic Study module to analyze a video of a skier competing. After calibrating the scale and pointing out the successive positions of the skier, the student analyzes the horizontal and vertical components of the speed then calculates the real speed by combining these two components to compare it to that displayed in the video.

Level:

Middle school

FizziQ

Author:

Duration (minutes) :

35

What students will do :

- Analyze a downhill skiing video using FizziQ kinematic tracking to determine the skier's speed
- Decompose velocity into horizontal and vertical components from the video analysis
- Reconstruct the true speed using the Pythagorean theorem: V = √(Vx² + Vy²)
- Compare the calculated speed with the broadcast speed value
- Understand the limitations of 2D video analysis for 3D motion

Scientific concepts:

- Velocity vectors
- Orthogonal component analysis
- Pythagorean theorem
- Board sports
- Visual Perspectives

Sensors:

- Camera (video recording for kinematic analysis)
- FizziQ Kinematics module (frame-by-frame position tracking)

What is required:

- Smartphone or tablet with the FizziQ application
- Olympic Games 'Downhill' video accessible via FizziQ resources
- FizziQ experience notebook

Experimental procedure:

Open FizziQ and navigate to the Kinematics module. Load the 'Downhill' skiing video from the FizziQ resource library.
Identify a reference scale in the video. Gate spacing, the skier's height, or course markings can be used.
Set the scale and define the coordinate system with x horizontal and y vertical.
Track the skier's position frame by frame through a section of the course where the speed appears approximately constant.
Collect at least 10-15 data points over 1-2 seconds of the run.
From the position data, calculate the horizontal velocity Vx (change in x per unit time) and the vertical velocity Vy (change in y per unit time).
Plot both Vx and Vy versus time. Note whether they are approximately constant or changing.
Calculate the total speed at each time step: V = √(Vx² + Vy²).
Convert V from m/s to km/h (multiply by 3.6) and compare with the speed displayed in the broadcast.
Discuss why the video-derived speed may differ from the broadcast speed. Consider the angle of the camera and perspective effects.
Analyze a section where the skier is accelerating or braking and observe how Vx and Vy change differently.
Create a summary table with time, Vx, Vy, V_total, and V_broadcast for each analyzed section.

Expected results:

Olympic downhill speeds typically range from 80-140 km/h (22-39 m/s). The video analysis may underestimate the true speed by 10-30% due to perspective effects: the camera view does not capture the component of velocity perpendicular to the image plane (toward or away from the camera). In sections where the skier moves parallel to the camera, the 2D analysis gives a better estimate. The horizontal component Vx dominates on gentle slopes, while Vy becomes significant on steep sections. Students should observe that the total speed V is always greater than either individual component, illustrating the Pythagorean relationship. Measurement precision is typically ±2-5 m/s due to pointing accuracy and frame rate limitations.

Scientific questions:

- Why is the total speed always greater than either the horizontal or vertical component alone?
- When is the 2D video analysis most accurate? When is it least accurate?
- What information is lost when a 3D motion is captured by a single camera?
- How could you improve the speed estimate using two cameras at different angles?
- Why does the skier's speed vary during the descent? What forces are at play?
- How do official timing systems measure the skier's speed during competition?

Scientific explanations:

The kinematic analysis of a skier in Olympic downhill allows us to explore several concepts of mechanics applied to high-level sport. The skier moves in three-dimensional space, but traditional video analysis captures this movement in two dimensions.

The video, generally taken from the side, makes it possible to measure two orthogonal components of the speed: Vx (horizontal) and Vy (vertical). The real speed V is the norm of the speed vector, calculated according to the Pythagorean theorem: V = √(Vx² + Vy²).

This distinction is crucial because the horizontal component alone significantly underestimates the skier's actual speed. For a slope of 30°, ignoring the vertical component can lead to an underestimate of 15%.

When pointing, the choice of a marker on the skier is decisive: the helmet generally provides a clearly visible point representative of the center of mass. Points like skis or feet follow more complex trajectories due to technical movements.

The discrepancies between the measured speed and that displayed in the video can be explained by several factors: 1) Perspective effects: the camera is never perfectly perpendicular to the trajectory; 2) The official radar measures instantaneous speed while video analysis calculates an average speed between two positions; 3) The accuracy of the scale calibration directly influences the results. The speeds in alpine skiing are impressive: 110-120 km/h downhill, and up to 160 km/h in speed skiing.

At these speeds, aerodynamic forces become preponderant, hence the importance of the "egg" position adopted by skiers to minimize drag.

Extension activities:

Frequently asked questions:

Q: My calculated speed is much lower than the broadcast speed. Is this expected?
R: Yes, if the skier is moving partly toward or away from the camera, that component of velocity is invisible in the 2D analysis. This systematically underestimates the true speed. The most accurate measurements come from sections where the skier moves perpendicular to the camera's line of sight.

Q: The position data is noisy and the velocities fluctuate wildly. How can I improve?
R: Use a larger interframe interval (track every 2-3 frames instead of every frame) to increase the spatial displacement between points, reducing the relative effect of pointing errors.

Q: Why does the skier's speed seem to change between frames even on a straight section?
R: Measurement noise from pointing imprecision creates apparent velocity fluctuations. Real speed changes (due to terrain, wind, or technique) also contribute. Average over several frames for a smoother estimate.

Q: How do the broadcast speed measurements work?
R: Official race timing uses electronic gates and radar systems. GPS-based systems provide continuous speed data. These are far more precise than video-based methods.