Billiards

This experiment allows students to observe and quantify the conservation of mechanical energy during a collision. It develops their ability to use video analysis to measure physical quantities and to interpret the gaps between theory and practice.

A billiard table provides one of the closest real-world approximations to the idealized collisions studied in physics textbooks. When one ball strikes another, the interaction lasts mere milliseconds, yet during that brief instant, momentum and energy are exchanged according to precise physical laws. In physics, collisions are classified as elastic (kinetic energy is conserved) or inelastic (some energy is lost to deformation, heat, or sound). Billiard ball collisions are often cited as examples of nearly elastic collisions, with restitution coefficients above 0.95, meaning that less than 5% of kinetic energy is lost during impact. However, even this small loss, combined with rolling friction on the felt surface and the transfer of translational energy into spin, means that energy conservation is never perfect in practice. This experiment challenges students to measure the velocities of billiard balls before and after a collision using FizziQ's video analysis tools, calculate the kinetic energies, and assess how closely the collision approaches the elastic ideal.

Learning objectives:

The student analyzes a video of a collision between two billiard balls using the FizziQ kinematic analysis module. After calibrating the scale, it tracks the position of the balls before and after impact to determine their respective speeds. By calculating the total kinetic energy of the system before and after the collision, it checks whether the mechanical energy is conserved and identifies the factors that could explain possible deviations.

Level:

High school

FizziQ

Author:

Duration (minutes) :

40

What students will do :

- Analyze a billiard ball collision using video-based kinematic tracking in FizziQ
- Measure the velocities of both balls before and after the collision
- Calculate the total kinetic energy before and after impact
- Determine the coefficient of restitution and characterize the elasticity of the collision
- Identify and quantify the factors responsible for energy dissipation

Scientific concepts:

- Conservation of mechanical energy, Elastic and inelastic collision, Kinematic analysis, Kinetic energy, Friction forces and energy dissipation

Sensors:

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

What is required:

- Smartphone with the FizziQ application, Billiard ball collision video (FizziQ library or personal video), Experiment book for calculations

Experimental procedure:

Open FizziQ and navigate to the Kinematics module. Load the billiard ball collision video from the FizziQ library, or import your own video.
Establish a scale using a known dimension in the video (e.g., the diameter of a billiard ball: 5.25 cm, or the width of a table pocket).
Set the coordinate origin and orient the x-axis along the initial direction of motion of the moving ball.
Begin tracking Ball 1 (the striking ball) by clicking on its center in each frame. Track at least 5 frames before and 5 frames after the collision.
Repeat the tracking process for Ball 2 (the target ball), using the same frames.
After completing the tracking, examine the position vs. time graphs for both balls in the FizziQ notebook.
Determine the velocity of Ball 1 before the collision (v₁) from the slope of the position-time graph before impact.
Determine the velocity of Ball 1 after the collision (v₁') from the slope after impact. It may be zero or reversed.
Determine the velocity of Ball 2 after the collision (v₂') from the slope of its position-time graph after impact. Note that Ball 2 was initially stationary (v₂ = 0).
Calculate the kinetic energies: Ec = ½mv² for each ball before and after the collision. Since billiard balls have equal masses, you can work with v² directly.
Compare the total kinetic energy before (½mv₁²) and after (½mv₁'² + ½mv₂'²) the collision. Calculate the percentage of energy lost.
Calculate the coefficient of restitution: e = (v₂' - v₁') / (v₁ - v₂). For a perfectly elastic collision, e = 1.

Expected results:

For a well-executed head-on collision between billiard balls, the striking ball should nearly stop (v₁' ≈ 0) while the target ball moves away at approximately the initial speed of the striker (v₂' ≈ v₁). The coefficient of restitution should be between 0.92 and 0.98. The total kinetic energy after the collision should be 85-96% of the initial kinetic energy, with losses due to sound generation (the characteristic click of a billiard collision), slight ball deformation, friction with the felt, and rotational energy transfer. For oblique (non-head-on) collisions, the analysis is more complex as both balls will have velocity components in two dimensions, but the total kinetic energy should still be approximately conserved. Pointing errors will introduce uncertainty of approximately ±5-10% in the velocity measurements.

Scientific questions:

- What physical factors explain the small percentage of energy lost during the collision?
- How would the result differ if the collision were perfectly head-on versus slightly off-center?
- Is momentum conserved in this collision? How could you verify this from your data?
- What role does the spin of the balls play in the collision outcome?
- How would the results change if you used balls of different masses (e.g., a snooker ball and a pool ball)?
- Why are billiard ball collisions considered nearly elastic while a car crash is highly inelastic?

Scientific explanations:

A perfectly elastic collision is characterized by the simultaneous conservation of kinetic energy and momentum. Kinetic energy is calculated by the formula Ec = ½mv², where m is the mass and v is the speed.

For a system of two balls, the total kinetic energy before the collision (Ec₁ + Ec₂) should equal that after the collision (Ec₁' + Ec₂'). In reality, even on a professional pool table, a collision is never perfectly elastic.

The restitution coefficient (e), defined by the ratio between the relative speed after and before impact, characterizes the elasticity of the shock. For billiard balls, e ≈ 0.92-0.98.

Several factors explain the loss of energy: temporary deformation of the balls upon impact, friction with the carpet, air resistance, and transfer of translational kinetic energy into rotational energy. Kinematic analysis by video allows precise measurements to be obtained without complex instruments.

Extension activities:

Frequently asked questions:

Q: The striking ball does not stop completely after the collision. Is this normal?
R: Yes, for real billiard collisions that are not perfectly head-on, the striking ball retains some velocity. Even for head-on impacts, the coefficient of restitution being slightly less than 1 means the striking ball may retain a small forward or backward velocity.

Q: My calculated energy loss is very large (more than 20%). What might be wrong?
R: Large apparent energy losses usually indicate velocity measurement errors. Verify your scale calibration, ensure you are tracking the ball center accurately, and use enough frames to get a reliable velocity estimate. Friction with the felt should cause only gradual deceleration, not large instantaneous energy loss.

Q: How can I track both balls if the collision happens very quickly?
R: Use slow-motion video if possible. In standard 30 fps video, the collision itself usually occurs within 1-2 frames, so you effectively measure velocities just before and just after impact using the surrounding frames. The instantaneous collision dynamics cannot be resolved at normal frame rates.

Q: Should I account for the balls rolling rather than sliding?
R: For this introductory analysis, treating the balls as point masses and considering only translational kinetic energy is sufficient. The rotational energy is typically a small fraction of the total, but accounting for it would bring the energy balance closer to perfect conservation.