Parabolic trajectory: basketball
This activity allows students to analyze the parabolic trajectory of a basketball during a shot. It develops skills in mathematical modeling and kinematic analysis.
Activity Summary
The student uses the FizziQ Kinematic Study module to analyze a basketball shooting video. After calibrating the scale and pointing the position of the ball frame by frame, the student obtains position data which he analyzes by applying mathematical interpolation to verify the parabolic nature of the trajectory and evaluate the precision of his pointing.
Introduction
Every basketball shot is an unconscious exercise in ballistics. The player must launch the ball at precisely the right angle and speed to arc through the air and fall through a hoop only 46 cm in diameter, positioned 3.05 meters above the floor. What makes this feat remarkable is that the trajectory, once the ball leaves the hand, is entirely determined by physics: gravity pulls the ball downward while its initial velocity carries it forward, tracing a graceful parabolic arc. This parabolic motion, first described mathematically by Galileo in the early 17th century, is one of the most fundamental predictions of Newtonian mechanics. Unlike a badminton shuttlecock, a basketball is heavy and compact enough that air resistance has negligible effect over the distances involved in a typical shot, making it an ideal projectile for verifying the theoretical parabola. Using FizziQ's kinematic analysis module, students can track the ball's position frame by frame in a video recording and quantitatively verify that the trajectory is indeed parabolic, while also estimating the initial speed and launch angle.
Learning Objectives
- Verify that the trajectory follows a parabolic path consistent with projectile motion theory
- Apply quadratic curve fitting to determine the equation of the trajectory
- Estimate the initial velocity and launch angle from the fitted parameters
- Evaluate the precision of manual frame-by-frame tracking through residual analysis
Scientific Concepts
Instruments and sensors
Scientific instruments
- Video (kinematics) analysis module
Sensors
- Camera
FizziQ Features
- Video resource library — provides the ready-to-use 'Basketball' shot video for the analysis.
- Curve fitting (quadratic interpolation) — fits y = ax² + bx + c to the tracked positions to extract the trajectory coefficients.
- Experiment notebook — displays the y vs. x position graph and the fit results for interpretation.
Required Materials
- - Smartphone or tablet with the FizziQ application - "Basketball" video accessible via FizziQ's resource library - FizziQ experiment notebook - Note: the protocol remains adaptable to any comparable video analysis tool.
Experimental Protocol
Open FizziQ and go to the **Kinematics** module. Load the **'Basketball'** video from the FizziQ resource library.
Identify a **reference scale** in the video. The basketball hoop height (3.05 m) or the diameter of the ball (24.3 cm for a standard men's ball) are good references.
Set the **scale** in FizziQ by clicking on two points of known separation in the video.
Define the **origin** of the coordinate system (e.g., the point where the ball leaves the player's hands) and orient the axes with x horizontal and y vertical.
Begin **frame-by-frame pointing**: click on the center of the ball in each successive frame. Start from the moment the ball leaves the shooter's hands.
Track the ball through its entire flight path until it reaches the basket or goes beyond the frame. Aim for at least **12-15 data points**.
Adjust the **interframe interval** (e.g., every 2 or 3 frames) to ensure the data points are well-separated spatially, reducing relative pointing errors.
After completing the pointing, view the **y vs. x position graph** in your FizziQ notebook. The shape should resemble a downward-opening parabola.
Use the FizziQ **curve fitting (interpolation)** tool to fit a **quadratic function** y = ax² + bx + c to the data.
Record the coefficients a, b, and c. From the coefficient a, calculate the initial horizontal velocity using v₀ₓ = √(-g / 2a), where g = 9.81 m/s².
Estimate the **launch angle** θ from the coefficient b, since b = tan(θ) for a trajectory starting at the origin.
Evaluate the quality of the fit by examining the **residuals** (differences between measured and fitted positions). Discuss sources of error.
Expected Results
The y vs. x graph should display a clear downward-opening parabola. The quadratic fit should yield a coefficient of determination (R²) above 0.99 for carefully tracked data. Typical values for the quadratic coefficient a range from -0.15 to -0.30, depending on the initial speed. From this, the initial horizontal velocity is typically 4-6 m/s for a standard shot, giving a total initial speed of 7-8 m/s at a launch angle of 45-55°. The residuals should be small (a few centimeters at most) and randomly distributed, indicating a good parabolic fit. Systematic deviations, if present, would suggest either air resistance effects or pointing inaccuracies. The largest source of error is typically the manual pointing precision, which depends on video resolution and ball visibility.
Scientific Questions
- How does the launch angle affect the shape of the parabola and the maximum height reached?
- From the equation y = ax² + bx + c, what physical information can you extract about the shot?
- What is the optimal launch angle for a three-point shot, and why is it not exactly 45°?
- How would the trajectory change if the ball had a significant spin? Does the Magnus effect play a role?
- What sources of error are most significant in this type of kinematic analysis?
Scientific Background
Basketball shooting perfectly illustrates the laws of Newtonian mechanics applied to the movement of projectiles. Once released, the ball follows an essentially parabolic trajectory, influenced primarily by two forces: gravity (constant, directed downward) and air resistance (generally negligible for a basketball over this distance).
The mathematical form of this trajectory is a parabola described by the parametric equations: x(t) = x₀ + v₀ₓt and y(t) = y₀ + v₀ᵧt - ½gt², where (x₀, y₀) is the initial position, (v₀ₓ, v₀ᵧ) the components of the initial velocity, g the acceleration of gravity (9.81 m/s²) and t the time. Quadratic interpolation of the curve y(x) gives an equation of the form y = ax² + bx + c, where the coefficient a is directly linked to g and to the initial horizontal speed by the relation a = -g/(2v₀ₓ²).
The kinematic analysis with FizziQ makes it possible to experimentally verify this relationship and to indirectly estimate the initial speed of the shot. To make a basket, the player must intuitively solve a complex ballistics problem, taking into account the distance to the basket, the height of the latter (3.05 m), and his own height.
For a shot at 6.75 m (three-point line), the optimal angle is approximately 45-50° and the necessary initial speed is approximately 7-8 m/s. Adjusting the interframe interval (176 ms) in the analysis helps optimize accuracy: intervals that are too short may capture positions that are too close, increasing the relative pointing error, while intervals that are too long may miss important details of the trajectory.
Extensions
- Compare the trajectories of shots from different distances and determine how the initial speed and angle change
- Analyze a free throw versus a three-pointer to compare launch parameters
- Use the kinematic data to calculate the ball's velocity and acceleration at each point and verify that the horizontal velocity remains approximately constant
- Attempt the analysis with a different sport (soccer throw-in, volleyball serve) and compare the trajectories
Frequently Asked Questions
My parabolic fit does not match the data well in some regions. What could be the cause?
Check your scale calibration carefully, as even a small error in the reference distance will distort all position measurements proportionally. Also verify that you are clicking consistently on the center of the ball in each frame. If some points are clearly outliers, you may re-point those specific frames.
How do I choose the right interframe interval for the analysis?
The ideal interval produces data points that are well-separated spatially (at least 5-10 cm apart in the image) but close enough to capture the curvature of the trajectory. For a typical basketball shot video at 30 fps, pointing every 2-3 frames usually works well.
The initial velocity I calculated seems unreasonable. How can I check it?
A typical basketball shot has an initial speed of 6-9 m/s. If your result is far outside this range, recheck your scale calibration and the units of your coefficients. Remember that a depends on both g and the horizontal velocity squared: a = -g/(2v₀ₓ²).
Can I use this method for a video I recorded myself?
Yes, but ensure the camera is positioned perpendicular to the plane of the trajectory (a side view), the camera is stationary during filming, and a scale reference is visible in the frame. Use the highest available frame rate for better resolution.
Detailed Description
The student uses the FizziQ Kinematic Study module to analyze a basketball shooting video. After calibrating the scale and pointing the position of the ball frame by frame, the student obtains position data which he analyzes by applying mathematical interpolation to verify the parabolic nature of the trajectory and evaluate the precision of his pointing. FizziQ provides the frame-by-frame video pointing tool, the built-in quadratic curve-fitting function, and the experiment notebook used to display and interpret the position graph and the fit results.
Ready to start?
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