How do we know how long the force is acting on an object?

[eprparadox]

Hey all, so I'm self studying and I came across this question:

A $2 kg$ cart, traveling on a horizontal air track with a speed of $3 m/s$, collides with a stationary $4 kg$ cart. The carts stick together. The impulse exerted by one cart on the other has a magnitude of:

A. $0$
B. $4 N \cdot s$
C. $6 N \cdot s$
D. $9 N \cdot s$
E.$12 N \cdot s$

So I know the answer is B and that's fine. We find the final speed of the combined system as

v_f = \frac{ 2kg \cdot 3m/s}{2 kg + 4 kg} = 1 m/s

The impulse is equal to the change in momentum of the stationary cart and so that is just equal to [$] 4kg \cdot 1 m/s = 4 kg \cdot m/s [/$]. That's all good.

My question is more of how we know how long a force is exerted. For example, if the objects didn't stick together, then the force exerted would be over the time that the two objects were touching each other.

In this case, is there a way to know this? Is is just as soon as the acceleration stops and we're moving at the combined final speed which is found by conservation of momentum?

 

[Dr.D]

There is no theoretical basis for determining the time duration of an impact force. Smaller magnitude acting for a slightly longer time has the same effect as greater magnitude acting for a shorter time.

It is possible to experimentally measure the time history of a particular impact force event, but that requires special equipment and is unique to that particular impact.

 

[Terry Bing]

In this case, is there a way to know this? Is is just as soon as the acceleration stops and we're moving at the combined final speed which is found by conservation of momentum?

I think so. When they are moving together after the collision, there is no acceleration (assuming friction is absent). So Newtons laws would tell you force on each mass should be zero.

 

[Dr.D]

I would agree with Terry that in the event of a perfectly plastic impact, such that the bodies travel on together, there is zero net force between them. But that does not mean that there is no force between them. Let me explain.

Consider a bullet shot and embedded in a ballistic pendulum. The bodies travel on together after impact. However, the material of the pendulum block is locally deformed where the bullet penetrates and lodges in the block. This local deformation causes a contact force to exist between the bullet at the block, even though the net force between them is zero.

 

[haruspex]

One can hypothesise a model for the impact process. E.g., likely to be an elastic deformation initially, in the sense that the force is proportional to the compression, but reaching a peak force and plateauing.

If the bodies stick together then the elastic phase will be short but not necessarily absent. The force may be proportional to compression during increasing compression and yet vanish as soon as it starts to decompress. Consider e.g. two blocks connected by a fragile spring. A body impacting one experiences increasing resistance as it pushes the first block along, then the spring breaks, so there is no rebound at the end. The impacted body may behave like that internally.