Estimation of g by kinematic study

This activity allows students to experimentally determine the value of the acceleration of gravity by video analysis. It concretizes the equation of parabolic motion using digital modeling tools.

The acceleration of gravity, denoted g, is one of the most fundamental constants in classical mechanics. Its value, approximately 9.81 m/s² at the Earth's surface, governs the fall of every object, from raindrops to satellites. Galileo was the first to measure g systematically in the early 17th century, using inclined planes to slow down falling objects enough to time them with a water clock. Today, geophysicists measure g with extraordinary precision (to nine significant figures) using atom interferometry and falling corner-cube retroreflectors. Between these extremes, video analysis offers a powerful and accessible method. By filming a ball in free flight and tracking its position frame by frame, students can extract the acceleration of gravity from the curvature of the parabolic trajectory. This experiment uses two independent methods, velocity-time analysis and position-time curve fitting, to determine g from the same video data, providing a valuable lesson in measurement methodology and data analysis.

Activity overview:

The student analyzes the parabolic trajectory of a ball using the FizziQ Kinematics module. After calibrating the scale of the video and pointing out the position of the ball on each image the student uses two complementary methods to determine g: first by analyzing the slope of the vertical velocity curve then by examining the coefficient of the quadratic term in the position equation.

Level:

High school

FizziQ

Author:

Duration (minutes) :

40

What students will do :

- Determine the acceleration of gravity experimentally using video-based kinematic analysis
- Apply two independent methods: linear fit of vertical velocity versus time, and quadratic fit of vertical position versus time
- Compare the two experimental values of g with each other and with the accepted value
- Evaluate measurement uncertainty and identify the dominant sources of error
- Understand the equations of uniformly accelerated motion through direct measurement

Scientific concepts:

- Parabolic movement
- Acceleration of gravity
- Equations of motion
- Data interpolation
- Kinematic analysis

Sensors:

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

Material needed:

- Smartphone or tablet with the FizziQ application
- 'Parable' video from the FizziQ library or personal video of an object in free fall
- FizziQ experience notebook

Experimental procedure:

Open FizziQ and navigate to the Kinematics module. Load the 'Parabola' video from the FizziQ resource library, or import your own video of a ball in free flight.
Identify a reference scale in the video (e.g., a meter stick, the height of a person, or a known distance between two objects). Set the scale in FizziQ.
Define the coordinate system with the y-axis pointing upward and the origin at a convenient location.
Track the ball frame by frame by clicking on its center in each successive frame. Collect at least 15-20 data points spanning the full trajectory.
After tracking, examine the y vs. t (vertical position versus time) graph. It should be a downward-opening parabola.
Method 1: Use FizziQ's curve fitting tool to fit a quadratic function y = y₀ + v₀t - ½gt² to the y(t) data. Extract the coefficient of the t² term and calculate g.
Method 2: Examine the v_y vs. t (vertical velocity versus time) graph. It should be a straight line with a negative slope.
Fit a straight line to the v_y(t) data. The slope equals -g. Record this value.
Compare the two estimates of g. They should agree within experimental uncertainty.
Also check the horizontal velocity v_x(t). It should be approximately constant throughout the flight, confirming negligible air resistance.
Calculate the percentage error of each estimate relative to the accepted value of 9.81 m/s².
Discuss the sources of uncertainty: scale calibration, pointing precision, video frame rate, air resistance, and timing accuracy.

Expected results:

Both methods should yield values of g between 9.0 and 10.5 m/s², with typical errors of 5-10% for careful measurements. The quadratic fit to y(t) tends to be more robust because it uses all the data simultaneously, while the velocity method amplifies pointing noise through numerical differentiation. The horizontal velocity should remain approximately constant (within ±10%), confirming that air resistance is negligible for a heavy ball over short distances. Common sources of systematic error include inaccurate scale calibration (which shifts g proportionally), parallax in the video, and motion blur at high speeds. The coefficient of determination for the quadratic fit should exceed 0.99.

Scientific questions:

- Why does the quadratic fit method generally give a more precise result than the velocity-slope method?
- What assumptions does this analysis make about air resistance? When would these assumptions break down?
- How would you modify this experiment to measure g more precisely?
- Why is the horizontal velocity approximately constant while the vertical velocity changes linearly?
- What factors cause the value of g to vary slightly from one location to another on Earth?
- How does the frame rate of the video affect the precision of the measurement?

Scientific explanations:

An object moving in the Earth's gravity field experiences a constant acceleration g ≈ 9.81 m/s² directed vertically downwards. This acceleration only affects the vertical component of the movement, the horizontal component remaining uniform in the absence of friction.

For vertical motion, the time equations are: y(t) = y₀ + v₀ₜt - ½gt² and v_y(t) = v₀ - gt. FizziQ's kinematic analysis tool allows you to experimentally verify these equations by tracking the position of an object frame by frame.

Two methods of estimating g are possible: 1) From the speed: v_y being a linear function of time, the slope of this line corresponds to -g. Linear interpolation provides this value directly.

2) From the position: y(t) being a quadratic function, the coefficient of the term t² is worth -g/2 in the interpolation y = at² + bt + c. Sources of error include: pointing inaccuracy, perspective effects if the camera is not perpendicular to the plane of motion, and the influence of air resistance which can slightly reduce the effective acceleration.

The use of two independent methods makes it possible to validate the consistency of the results. This experiment perfectly illustrates the universality of the law of gravitation: all bodies fall with the same acceleration regardless of their mass, a principle discovered by Galileo in the 17th century.

Extension activities:

Frequently asked questions:

Q: My value of g is significantly different from 9.81 m/s². What are the most likely causes?
R: The most common cause is an error in scale calibration. A 5% error in the reference distance translates directly to a 5% error in g from the quadratic fit (and 10% from the velocity fit). Double-check your reference distance and ensure the camera was perpendicular to the plane of motion.

Q: The vertical velocity graph is very noisy. How can I improve it?
R: Velocity is calculated by differentiating position, which amplifies noise. Use more data points, track the ball more precisely, and consider using a larger interframe interval so that position differences are larger relative to the pointing error.

Q: Can I use this method with a smartphone camera instead of a professional video?
R: Yes, but ensure the smartphone is fixed on a tripod and records at a known, stable frame rate (30 or 60 fps). The higher the frame rate, the better the time resolution.

Q: Why does Method 1 (quadratic fit) work better than Method 2 (velocity slope)?
R: The quadratic fit uses the raw position data directly, while the velocity is derived from differences between consecutive positions. Differencing amplifies small errors, making the velocity data noisier than the position data.