# Maximum deformation and velocity

When two bodies A and B are travelling in one direction with velocity of A greater than the velocity of B. When the two bodies collide, deformation takes place. My question is at the time of maximum deformation a) Why is the normal force maximum b) Why do the velocities become equal for a very small amount of time? I know in the case of spring at the time of maximum compression, the velocities become equal because if they are not equal, then more compression will take place.

• Instinctively, maximum deformation would not occur at a maximum normal force for all systems. If I create a metamaterial consisting of a steel plate with ballistics gel behind it, and fire a bullet at it, the maximum force clearly occurs while the bullet is going through the steel plate, but the maximum deformation is at the end of the flight through the gel. Commented Sep 25, 2017 at 18:58
• Yes, well for uniform bodies, then. Anyway you seem to have answered that question yourself I'm not sure if you have any remaining dilema? An elastically deformable body is a spring. Commented Sep 25, 2017 at 20:03
• Commented Sep 25, 2017 at 23:01

Let's look at the two parts of your scenario:

At the time of maximum deformation, the normal force is maximum

An elastic body distorts under the influence of stress; for elastic solids, the strain (degree of distortion) is proportional to the stress. The greatest strain will happen at the time of greatest stress - and thus, when the (normal) force is maximum.

The second part now follows.

Velocities become equal for a short amount of time

If the objects are still moving towards each other at the time of greatest normal force, they will be more deformed at a later time; but greater deformation means greater force - so if they are still moving towards each other, the force has not yet reached a maximum. Conversely, once they are moving away from each other, the deformation becomes less, and the force is reducing. Between "moving towards each other" and "moving away from each other", there is a moment in time when they are "neither moving towards, nor away from each other". That's the moment of greatest deformation, greatest normal force, and equal velocity.

Note - the above is true for elastic deformation only. If you have inelastic behavior, the force may be greatest before the deformation reaches a maximum (for example, if you collide a glass sphere with a solid metal block, the force is greatest when the sphere breaks; but at that point it may still be moving toward the block).