Theory

In the theory of one-dimensional and two-dimensional collisions, understanding the fundamental principles of conservation of momentum and energy is key. Here's a detailed description of the theoretical framework for both types of collisions:

One-Dimensional Collision Theory

In one-dimensional (1D) collisions, all objects move along a single straight line. Collisions can be classified as elastic or inelastic, based on whether kinetic energy is conserved.

1. Conservation of Momentum

The total linear momentum in an isolated system (without external forces) remains constant before and after the collision. For two objects with masses m1​ and m2​ and initial velocities v1i and v2i, the conservation of momentum is expressed as:

where v1f​ and v2f are the final velocities after the collision.

2. Elastic Collision

In an elastic collision, both momentum and kinetic energy are conserved. The conservation of kinetic energy is given by:

For elastic collisions, both equations (momentum and kinetic energy conservation) can be solved to find the final velocities v1f and v2f.

3. Inelastic Collision

In an inelastic collision, only momentum is conserved. Kinetic energy is not conserved because some of it is transformed into other forms of energy, such as heat or sound. A special case is a perfectly inelastic collision, where the two objects stick together after the collision. The equation for momentum conservation still holds:

where vf is the final velocity of the combined mass.

Two-Dimensional Collision Theory

In two-dimensional (2D) collisions, the motion occurs in a plane, and both the x- and y-components of momentum must be considered separately. Again, momentum is always conserved, and energy conservation depends on whether the collision is elastic or inelastic.

1. Conservation of Momentum in Two Dimensions

For two objects with masses m1​ and m2, initial velocities v1i and v2i​, and final velocities v1f​ and v2f​, the conservation of momentum in the x- and y-directions is given by:

• In the x-direction:

In the y-direction:

The velocities are broken down into their respective components in the x and y directions using basic trigonometry.

2. Elastic Two-Dimensional Collision

For an elastic collision, in addition to momentum conservation in both directions, the total kinetic energy is conserved:

3. Inelastic Two-Dimensional Collision

In the case of inelastic collisions, only momentum is conserved, and kinetic energy is lost to other forms. For perfectly inelastic collisions, where the objects stick together, the final velocity vector is the same for both objects:

• In the x-direction:

• In the y-direction:

where vf is the magnitude of the final velocity, and θf​ is its direction.

ImageJ Analysis

 When using ImageJ to analyze the collision, the video frames are examined to extract position data over time for both objects. Plotting the spatial position against time makes it possible to verify the theoretical predictions for velocities and confirm the conservation laws (momentum and kinetic energy in the elastic case).

 This combination of simulation using PhET and video analysis through ImageJ offers a comprehensive way to study the dynamics of collisions in one and two dimensions.

ImageJ Analysis

 When using ImageJ to analyze the collision, the video frames are examined to extract position data over time for both objects. Plotting the spatial position against time makes it possible to verify the theoretical predictions for velocities and confirm the conservation laws (momentum and kinetic energy in the elastic case).

 This combination of simulation using PhET and video analysis through ImageJ offers a comprehensive way to study the dynamics of collisions in one and two dimensions.