Inelastic collision

Collision in which energy is lost to heat

A bouncing ball captured with a stroboscopic flash at 25 images per second. Each impact of the ball is inelastic, meaning that energy dissipates at each bounce. Ignoring air resistance, the square root of the ratio of the height of one bounce to that of the preceding bounce gives the coefficient of restitution for the ball/surface impact.

An inelastic collision, in contrast to an elastic collision, is a collision in which kinetic energy is not conserved due to the action of internal friction.

In collisions of macroscopic bodies, some kinetic energy is turned into vibrational energy of the atoms, causing a heating effect, and the bodies are deformed.

The molecules of a gas or liquid rarely experience perfectly elastic collisions because kinetic energy is exchanged between the molecules' translational motion and their internal degrees of freedom with each collision. At any one instant, half the collisions are – to a varying extent – inelastic (the pair possesses less kinetic energy after the collision than before), and half could be described as “super-elastic” (possessing more kinetic energy after the collision than before). Averaged across an entire sample, molecular collisions are elastic. ^[1]

Although inelastic collisions do not conserve kinetic energy, they do obey conservation of momentum.^[2] Simple ballistic pendulum problems obey the conservation of kinetic energy only when the block swings to its largest angle.

In nuclear physics, an inelastic collision is one in which the incoming particle causes the nucleus it strikes to become excited or to break up. Deep inelastic scattering is a method of probing the structure of subatomic particles in much the same way as Rutherford probed the inside of the atom (see Rutherford scattering). Such experiments were performed on protons in the late 1960s using high-energy electrons at the Stanford Linear Accelerator (SLAC). As in Rutherford scattering, deep inelastic scattering of electrons by proton targets revealed that most of the incident electrons interact very little and pass straight through, with only a small number bouncing back. This indicates that the charge in the proton is concentrated in small lumps, reminiscent of Rutherford's discovery that the positive charge in an atom is concentrated at the nucleus. However, in the case of the proton, the evidence suggested three distinct concentrations of charge (quarks) and not one.

Formula

The formula for the velocities after a one-dimensional collision is: $${\displaystyle {\begin{aligned}v_{a}&={\frac {C_{R}m_{b}(u_{b}-u_{a})+m_{a}u_{a}+m_{b}u_{b}}{m_{a}+m_{b}}}\\v_{b}&={\frac {C_{R}m_{a}(u_{a}-u_{b})+m_{a}u_{a}+m_{b}u_{b}}{m_{a}+m_{b}}}\end{aligned}}}$$

where

• v_a is the final velocity of the first object after impact
• v_b is the final velocity of the second object after impact
• u_a is the initial velocity of the first object before impact
• u_b is the initial velocity of the second object before impact
• m_a is the mass of the first object
• m_b is the mass of the second object
• C_R is the coefficient of restitution; if it is 1 we have an elastic collision; if it is 0 we have a perfectly inelastic collision, see below.

In a center of momentum frame the formulas reduce to:

$${\displaystyle {\begin{aligned}v_{a}&=-C_{R}u_{a}\\v_{b}&=-C_{R}u_{b}\end{aligned}}}$$

For two- and three-dimensional collisions the velocities in these formulas are the components perpendicular to the tangent line/plane at the point of contact.

If assuming the objects are not rotating before or after the collision, the normal impulse is:

$${\displaystyle J_{n}={\frac {m_{a}m_{b}}{m_{a}+m_{b}}}(1+C_{R})({\vec {u_{b}}}-{\vec {u_{a}}})\cdot {\vec {n}}}$$

where ${\displaystyle {\vec {n}}}$ is the normal vector.

Assuming no friction, this gives the velocity updates:

$${\displaystyle {\begin{aligned}\Delta {\vec {v_{a}}}&={\frac {J_{n}}{m_{a}}}{\vec {n}}\\\Delta {\vec {v_{b}}}&=-{\frac {J_{n}}{m_{b}}}{\vec {n}}\end{aligned}}}$$

Perfectly inelastic collision

A completely inelastic collision between equal masses

A perfectly inelastic collision occurs when the maximum amount of kinetic energy of a system is lost. In a perfectly inelastic collision, i.e., a zero coefficient of restitution, the colliding particles stick together. In such a collision, kinetic energy is lost by bonding the two bodies together. This bonding energy usually results in a maximum kinetic energy loss of the system. It is necessary to consider conservation of momentum: (Note: In the sliding block example above, momentum of the two body system is only conserved if the surface has zero friction. With friction, momentum of the two bodies is transferred to the surface that the two bodies are sliding upon. Similarly, if there is air resistance, the momentum of the bodies can be transferred to the air.) The equation below holds true for the two-body (Body A, Body B) system collision in the example above. In this example, momentum of the system is conserved because there is no friction between the sliding bodies and the surface. $${\displaystyle m_{a}u_{a}+m_{b}u_{b}=\left(m_{a}+m_{b}\right)v}$$where v is the final velocity, which is hence given by $${\displaystyle v={\frac {m_{a}u_{a}+m_{b}u_{b}}{m_{a}+m_{b}}}}$$The reduction of total kinetic energy is equal to the total kinetic energy before the collision in a center of momentum frame with respect to the system of two particles, because in such a frame the kinetic energy after the collision is zero. In this frame most of the kinetic energy before the collision is that of the particle with the smaller mass. In another frame, in addition to the reduction of kinetic energy there may be a transfer of kinetic energy from one particle to the other; the fact that this depends on the frame shows how relative this is. The change in kinetic energy is hence: $${\displaystyle \Delta KE={1 \over 2}\mu u_{\rm {rel}}^{2}={\frac {1}{2}}{\frac {m_{a}m_{b}}{m_{a}+m_{b}}}|u_{a}-u_{b}|^{2}}$$

where μ is the reduced mass and u_rel is the relative velocity of the bodies before collision. With time reversed we have the situation of two objects pushed away from each other, e.g. shooting a projectile, or a rocket applying thrust (compare the derivation of the Tsiolkovsky rocket equation).

Partially inelastic collisions

Partially inelastic collisions are the most common form of collisions in the real world. In this type of collision, the objects involved in the collisions do not stick, but some kinetic energy is still lost. Friction, sound and heat are some ways the kinetic energy can be lost through partial inelastic collisions.

See also

• Collision
• Elastic collision
• Coefficient of restitution

References

1. Hernandez, Hugo (2023). "Confusion and Illusions in Collision Theory". ForsChem Research Reports. 8. doi:10.13140/RG.2.2.24913.10088. Retrieved 25 August 2024 – via ResearchGate.
2. Ferdinand Beer Jr. and E. Russell Johnston (1996). Vector equations for engineers: Dynamics (Sixth ed.). McGraw Hill. pp. 794–797. ISBN 978-0070053663. If the sum of the external forces is zero ... the total momentum of the particles is conserved. In the general case of impact, i.e., when e is not equal to 1, the total energy of the particles is not conserved.