Elastic Collisions

3 Key excerpts on "Elastic Collisions"

• 

  eBook - ePub

  Biomechanics of Human Motion

  Applications in the Martial Arts, Second Edition

  • Emeric Arus, Ph.D.(Authors)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  In this case, we speak about elasticity of the object(s). A collision in which the total kinetic energy after the collision is less than that before the collision is called an inelastic collision. To be more explicit, the body or bodies will change their physical shape. Example includes a car crash or when two bodies stick together after the collision, for example, a sticky material thrown against the wall. A collision in which the total kinetic energy of a system will remain the same after the collision is called elastic collision. To be more explicit, the bodies will deflect each other with no physical change of shape. An example is when two billiard balls collide. The body has the tendency to return to its normal shape once it has been deformed, that is, its elasticity differs from one body to another. Some return very quickly to their original shape, while others do so less quickly. Because there is no way of directly calculating the elasticity of a body, it is necessary to rely on the different experiment results to predict the outcome of any given impact. Newton formulated an empirical law, Newton’s law of impact, which states that if two bodies move toward one another along the same straight line, the difference between their velocities before the impact is proportional to the difference between their velocities after the impact. In order to correctly calculate the velocities before and after the impact, there is a term coefficient of restitution, which must be used in our calculation. The coefficient of restitution (e) or COR is an indicator of elasticity of an object reflecting the ability of the object to return to its original shape once deformed, measured by the ratio of the impulse of rebound to the impulse of impact. This coefficient has a value between “1” and “0.” The value of “1” indicates an elastic collision. The value toward “0” indicates an inelastic collision

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  eBook - ePub

  Automotive Accident Reconstruction

  Practices and Principles, Second Edition

  • Donald E. Struble, John D. Struble(Authors)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  ε = 0; some (but not all) of the kinetic energy is lost. Solving for the closing velocity, we have

  V cl

  =

  2

  1 −

  ε 2

  m 1

  +

  m 2

  m 1

  m 2

  Δ KE

  .

  (19.18)

  One of the great conservation laws discussed in Chapter 1 is the conservation of energy. We can see, from Equation (19.17), that in general there is a loss of kinetic energy in a vehicle collision. Where did the energy go? In physics, “heat” and maybe “light” are the answers that turn up among the usual suspects. The same can be said for vehicle crashes, but heat and light are not the major players. The main destination of dissipated kinetic energy is, in fact, crush energy, which is obvious from looking at the vehicle(s) postcrash. The Law of Energy Conservation can be expressed simply as

  Δ KE =

  CE tot

  ,

  (19.19)

  where CEtot is the total crush energy among the various vehicles involved. Substitution into Equation (19.18) results in

  V cl

  =

  2

  1 −

  ε 2

  m 1

  +

  m 2

  m 1

  m 2

  CE tot

  .

  (19.20)

  This simple equation has important implications for all manners of reconstructions—not just uniaxial crashes. First of all, the closing velocity is directly related but not proportional to the total crush energy. This is a classic “damage only” solution, which is devoid of scene information except for restitution coefficient (which is, of course, related to the vehicle exit velocities). A “damage-only” reconstruction may be appealing in a case where the scene information is missing or nonexistent. However, crush energy knowledge about only one vehicle will not suffice; one needs to know all crush energies to effect a damage-only solution. This is an important fact to keep in mind when planning how to reconstruct an accident.

  Second, a reconstruction solution may be checked for reasonableness by looking at the reconstructed closing velocity. If that quantity is out of whack, the likely culprit is the calculation of one or both of the vehicle crush energies. It is another way the mathematics whispers in the ear of the reconstructionist as to where the truth resides.

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  eBook - ePub

  Mechanical Vibration

  Analysis, Uncertainties, and Control, Fourth Edition

  • Haym Benaroya, Mark Nagurka, Seon Han(Authors)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  The coefficient of restitution can be shown to be a measure of the kinetic energy lost in the collision. In a perfectly elastic collision, there is no loss and e = 1. In an inelastic collision, 0 < e < 1. Some kinetic energy is transformed into deformation of the material, heat, sound, and other forms of energy, and is therefore unavailable. This loss is represented in the model by damping, c. The denominator of Equation 4.48 is simply the speed of the ball prior to the first contact, v 0, and the numerator is the rebound or post‐impact speed of the ball, v 1. The latter can be found by differentiating Equation 4.45 and imposing the assumption mg / k ≪ | v 0 / ω d |, or alternatively, differentiating Equation 4.46 directly to give an expression for the velocity, x ˙ (t) = c v 0 2 m ω d exp (- c 2 m t) sin ω d t - v 0 exp (- c 2 m t) cos ω d t, and then substituting t = ΔT with Equation 4.47 to give the rebound. velocity, v 1 = x ˙ (Δ T) = v 0 exp (- c π 2 m ω d) = v 0 exp (- c Δ T 2 m) From Equation 4.48, the coefficient of restitution can be written simply as (4.49) e = exp (- c Δ T 2 m). For fixed m and c, the shorter the contact time, ΔT, the larger the coefficient of restitution, e, and vice versa. By manipulating Equations 4.43, 4.47, and 4.49, the stiffness and viscous damping can be written, respectively,. as (4.50) k = m (π Δ T) 2 1 + (ln e π) 2 ] (4.51) c = - 2 m Δ T ln e. Thus, a shorter contact time, ΔT, corresponds to both a higher stiffness, k, and damping, c. In addition, as e increases, there is a negligible change in k and a reduction in c. Assuming k, c, and e are constant (independent of the velocity v 0), ΔT will be constant for each contact since ω d depends only on the system parameters k, c, and m. 4.11.3 Natural Frequency &Damping Ratio The undamped natural frequency, ω n = k / m, can be expressed from Equation 4.50 as (4.52) ω n = π Δ T 1 + (ln e π) 2, and is a function of the contact time, and the coefficient of restitution

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