Impact Analysis on a Collision: Understanding Force, Deceleration, and Energy

When someone runs into a wall at a speed (x) and is brought to a stop in a time (y), the impact can be analyzed in terms of the kinetic energy and the forces involved. This article delves into the physics behind such collisions, exploring how the time interval of the impact affects the force and energy involved.

Kinetic Energy and Deceleration

The kinetic energy ((KE)) of the person before the impact can be calculated using the formula: [ KE frac{1}{2}mv^2 ] where (m) is the mass of the person and (v) is the speed in this case (x).

When the person hits the wall and comes to a stop, the deceleration can be calculated using the formula: [ a frac{Delta v}{Delta t} frac{0 - x}{y} -frac{x}{y} ]

The force ((F)) exerted during the impact can be found using Newton's second law: [ F ma m left(-frac{x}{y}right) -frac{mx}{y} ]

Impact and Damage

As the time interval (y) approaches zero, the deceleration becomes very large, and the force of impact increases. The impulse change in momentum can be calculated as: [ text{Impulse} F cdot y -frac{mx}{y} cdot y -mx ]

The negative sign indicates a change in direction but in terms of magnitude, it reflects the force exerted during the collision.

Convergence to Energy

As (y) approaches zero, the forces involved become infinitely large, but the total energy involved in the impact remains finite and is equal to the initial kinetic energy. The energy dissipated during the impact, which relates to the damage caused, is indeed converging to the kinetic energy of the system. Thus, regardless of how quickly the impact occurs, the energy available for damage is still determined by the initial kinetic energy: [ text{Energy dissipated} approx KE frac{1}{2}mv^2 ]

Analysis Using Basic Physics

If the (x) speed stays constant while the (y) time is the variable, we can use basic physics to shed some light on your question. An object with mass (m) moving with a velocity (v) has 'momentum' (mv), a very important quantity.
When this 'body' hits the wall, its velocity gets changed to zero.

Newton's Second Law of Motion tells us that there is a "force" whose sole job is to change momentum! Mathematically, each change in momentum must be exactly equal to the amount of force that is acting multiplied by the amount of time that the force acts.

For your situation, the change in momentum will always be the same: since the starting speed, ending speed, and mass are the same, the momentum change is constant. Therefore, the product of force times time will stay constant.

Bottom Line: If you can stretch out the time of your collision with the wall, you will have a less forceful, gentler collision.

I have ignored 'energy' because it is much more complicated and didn't seem very relevant to your situation. However, a teaser is that if you can stretch out the distance over which the collision occurs, energy calculations can be important.

This paragraph is actually how Newton wrote it. People have rearranged things as (F ma), but I have a STRONG preference for the momentum interpretation.

Understanding the forces and energy involved in collisions is crucial for assessing the damage and potential harm. By stretching out the time of impact, the peak forces can be reduced, which is a significant factor in mitigating injuries and damage.

Conclusion

In summary, while the forces during the impact may become very large as (y) approaches zero, the total energy available for damage remains determined by the initial kinetic energy of the person. This energy ultimately relates to the potential for damage upon collision with the wall. By focusing on the initial kinetic energy and the duration of the collision, we can better understand and mitigate the impact of such events.