A plastic collision, also known as a perfectly inelastic collision, occurs when two objects collide and stick together after impact, moving as a single combined object with the same final velocity. This represents the maximum possible kinetic energy loss while still conserving momentum. Despite the name "plastic," this term applies to any collision where objects remain joined after impact, whether through deformation, adhesion, or mechanical coupling.
Conservation Laws:
A plastic collision, also known as a perfectly inelastic collision, is defined by the conservation of linear momentum and the maximum possible loss of kinetic energy. The key characteristic is that the colliding bodies stick together and move with a single, common velocity after impact.
| Property | Details |
|---|---|
| Conservation Laws | Linear momentum is conserved. Kinetic energy is not conserved; it is maximally converted into other forms (heat, sound, deformation). |
| Key Feature | The colliding objects coalesce or stick together, moving as a single mass with a common final velocity after the collision. |
| Relevant Quantities | Mass (kg), Velocity (m/s), and Momentum (kg⋅m/s) are the primary quantities. The change in Kinetic Energy (Joules) is also a critical outcome. |
| Vector Nature | Momentum and velocity are vector quantities. In collisions occurring in more than one dimension, momentum must be conserved independently along each axis. |
| Dimensional Formula | The dimensional formula for the conserved quantity, momentum, is [M][L][T]⁻¹. The dimensional formula for kinetic energy is [M][L]²[T]⁻². |
| Symbol | Quantity | SI Unit | Description |
|---|---|---|---|
| \( v_f \) | Final Velocity | m/s | The common velocity of both objects after they stick together. |
| \( m_1, m_2 \) | Mass | kg | The masses of object 1 and object 2, respectively. |
| \( v_1, v_2 \) | Initial Velocity | m/s | The initial velocities of object 1 and object 2 before the collision. |
| \( p_i, p_f \) | Momentum | kg⋅m/s | The initial and final momentum of the system. |
| \( KE_i, KE_f \) | Kinetic Energy | J | The initial and final kinetic energy of the system. |
| \( \Delta KE \) | Change in Kinetic Energy | J | The amount of kinetic energy lost during the collision, converted to other forms. |
The formula for the final velocity in a plastic collision is derived directly from the principle of conservation of linear momentum, which states that the total momentum of an isolated system remains constant.
Step 1: Define the total initial momentum of the system.
The total momentum before the collision is the vector sum of the individual momenta of the two objects.
Step 2: Define the total final momentum of the system.
After the collision, the two objects stick together, forming a single combined mass \( (m_1 + m_2) \) that moves with a common final velocity \( v_f \).
Step 3: Apply the conservation of momentum.
Equate the total initial momentum to the total final momentum.
Step 4: Solve for the final velocity \( v_f \).
Isolate \( v_f \) by dividing both sides by the total mass \( (m_1 + m_2) \).
Plastic collisions can be classified based on the geometry of the impact and the initial state of motion of the colliding objects. Each case is solved by applying the principle of conservation of momentum.
| Type / Case | Description | When to Use |
|---|---|---|
| One-Dimensional (Head-On) | The initial velocities of both objects lie along the same straight line. The combined mass continues to move along this line. | Used for problems where motion is constrained to a single axis, such as two carts on a track colliding and sticking together. |
| Two-Dimensional (Oblique) | The initial velocities of the objects are not collinear. The combined mass moves off at an angle to the initial paths. | Applicable for analyzing glancing collisions where objects fuse, requiring momentum to be conserved in two separate perpendicular components (e.g., x and y). |
| Stationary Target | A common scenario where one of the objects is at rest (initial velocity is zero) before being struck by the other. | Simplifies the initial momentum calculation. A classic example is the ballistic pendulum, where a projectile embeds itself in a stationary block. |
| Explosions (Reverse Collision) | While not a collision, the mathematics are identical but reversed in time. A single object breaks apart into multiple pieces. | Used to analyze situations where internal forces push objects apart from an initial state of rest, like the recoil of a cannon when a shell is fired. |
Used in crash test analysis, designing vehicle crumple zones, collision reconstruction, and developing safety systems.
Applied to studies of bullet penetration, body armor testing, crime scene reconstruction, and ammunition design.
Essential for impact testing, material deformation studies, designing energy-absorbing composites, and failure analysis.
Used in helmet design, development of protective equipment, injury prevention research, and setting rules for contact sports.
Principles are used in the design of shock absorbers, protective packaging, machinery safety guards, and vibration dampening systems.
Important for designing spacecraft shielding against micrometeoroids, creating soft landing systems, and analyzing debris impacts.
Automobile Crumple Zones
The front and rear sections of a car are designed to crumple during a severe collision. This intentional deformation creates a plastic collision, extending the impact time and converting the car's kinetic energy into the work of bending metal, which protects the passengers in the rigid safety cell.
Modeling Clay
When two balls of modeling clay are thrown at each other, they stick together and deform upon impact. This is a classic, tangible example of a perfectly inelastic collision where kinetic energy is lost to the work of deforming the clay and a small amount of heat.
Catching a Football
When a football player catches a pass, the ball and the player's hands (and body) move together as one unit immediately after the catch. This is an approximation of a plastic collision. The player's arms and body absorb the ball's kinetic energy to bring it to a stop relative to their body.
| Quantity | Symbol | SI Unit | Dimensions |
|---|---|---|---|
| Mass | \( m \) | kilogram (kg) | \( [M] \) |
| Velocity | \( v \) | meter per second (m/s) | \( [L][T]^{-1} \) |
| Momentum | \( p \) | kilogram-meter per second (kg⋅m/s) | \( [M][L][T]^{-1} \) |
| Kinetic Energy | \( KE \) | Joule (J) | \( [M][L]^2[T]^{-2} \) |
Dimensional Analysis of the Final Velocity Formula:
We can check the formula \( v_f = \frac{m_1v_1 + m_2v_2}{m_1 + m_2} \) for dimensional consistency. The dimensions of the numerator are those of momentum, \( [M][L][T]^{-1} \), and the dimensions of the denominator are those of mass, \( [M] \). Therefore:
The resulting dimensions \( [L][T]^{-1} \) are those of velocity, confirming the formula is dimensionally correct.
The formula, derived from the conservation of momentum, is m1*v1i + m2*v2i = (m1 + m2)*vf. It calculates the final velocity (vf) of two objects after they collide and stick together. The total initial momentum of the system is set equal to the final momentum of the single, combined mass.
In the equation, m1 and m2 represent the masses of the two objects in kilograms (kg). The terms v1i and v2i are their initial velocities, and vf is their common final velocity after sticking together, all measured in meters per second (m/s). Remember that velocity is a vector, so direction matters.
This formula is used in any situation where two objects collide and move as a single unit afterward, such as a meteorite striking the Earth or two clay balls hitting and merging. It is applied to find the final velocity of the combined object when the initial masses and velocities are known. This is also known as a perfectly inelastic collision.
A frequent error is assuming that kinetic energy is conserved. In a plastic collision, kinetic energy is maximally lost, converted into heat, sound, and deformation. Only momentum is conserved, so you must not attempt to equate the initial and final kinetic energies.
Automotive engineers apply plastic collision principles to design vehicle crumple zones. These zones are meant to deform and 'stick' during a collision, absorbing kinetic energy and extending the impact time. This process reduces the force exerted on passengers, making the vehicle safer.
A plastic collision is a perfect illustration of a system where momentum is conserved but kinetic energy is not. The total momentum before the collision equals the total momentum after. However, the total kinetic energy after the collision is always less than the initial kinetic energy, as some of it is transformed into other energy forms.