Confusion with multiple impulses

[GKRM]

Suppose I have a body of mass M kept on a horizontal smooth surface. I strike that particle with some force ,providing an impulse J_1. The applied force is in vertically downward direction. Suppose that the body deforms in this process and it doesn't rebound. What I want to know is, whether the impulsive normal starts to act as soon as I apply a force or is there a time lag between the two? In other words I can look at this situation from two angles:-

1. Either I view it as a series of impulses imparted to the body. Which means that the particle when struck by external means, gains some velocity and then it's again acted upon by an impulsive normal which brings it to halt.
2. Or impulsive normal starts to act as soon as the external impulse is acting. That is the body doesn't gain any velocity at any instant.
By velocity I mean the velocity of the centre of mass of the body.
Which approach should I consider?
What should I consider for coefficient of restitution equation.. should I consider the final velocities after all impulses have acted on a body or should I view it as a step by step process.
Thanks in advance.

 

[A.T.]

Suppose I have a body of mass M kept on a horizontal smooth surface. I strike that particle with some force ,providing an impulse J_1. The applied force is in vertically downward direction. Suppose that the body deforms in this process and it doesn't rebound. What I want to know is, whether the impulsive normal starts to act as soon as I apply a force or is there a time lag between the two? In other words I can look at this situation from two angles:-

1. Either I view it as a series of impulses imparted to the body. Which means that the particle when struck by external means, gains some velocity and then it's again acted upon by an impulsive normal which brings it to halt.
2. Or impulsive normal starts to act as soon as the external impulse is acting. That is the body doesn't gain any velocity at any instant.
By velocity I mean the velocity of the centre of mass of the body.
Which approach should I consider?

Since the body deforms, the center of mass will move, so it's option 1. In general, the lag between the down force on top, and the increased up force on bottom will be determined by the speed of sound in the material.

 

[sophiecentaur]

Striking a 'real' body will launch a longitudinal (mostly) wave through the material. This will travel at the speed of sound.
c=√(K_s/ρ
where K_s is the bulk modulus and ρ is the density.
What actually happens to the CM of the body will depend on the nature of the surface it is alien on, the dimensions and the nature of the Impulse(s). It's not clear what the application is but, for instance, you could be knocking a nail in with a hammer or a riveting gun (multiple fast impacts). The penetration of the nail could be optimised by getting the repetition rate right so that the energy transfer into the target material is as large as possible.
Do you have any more details about your query?

 

[GKRM]

Well for my concerns you can assume that the body deforms and nothing pierces the body. The body can soon reform to gain it's original shape making the collision elastic.
My concerns are regarding whether the impulsive normal, as mentioned above start to act as soon as I initiate a contact between the body?
You can also assume the the body is spherical in shape, if that's any bit of information to you.

 

[sophiecentaur]

Well for my concerns you can assume that the body deforms and nothing pierces the body. The body can soon reform to gain it's original shape making the collision elastic.

So the 'table' is totally rigid and the collision is perfectly elastic? OK
If the period over which the force acts is long then the body will deform in a quasi static way. If it is uniform, its CM will move half as far as the place where the force is applied - much the same as stretching a wire. Once the impulse period is comparable with the time for the wave to travel through the body then the reflected wave will be relevant and it gets more complicated. A rapid set of impulses could set up a standing wave and near resonance the motion of the CM could be greater than the motion of the face that's struck.
We could take it further and say that, after a single impulse against a lossless material, the energy imparted would end up as Kinetic Energy as the object bounces away from the table. The body would also 'ring' as it constantly gets longer and shorter at its resonant frequency.
PS It might be easier to replace your 'body' with an ideal coil spring and analyse that situation first.

 

[Nidum]