I How do we analyze collisions involving accelerating objects?

[MattGeo]

I am not sure why it never occurred to me before despite actually having taken an advanced classical mechanics course in college, but how do we treat a collision where the objects involved are actually accelerating?

In the case where colliding objects move at constant velocity it is standard and very convenient to use conservation of momentum and conservation of energy in tandem to analyze how the colliding objects behave after the interaction.

Say you have a block with a net external force on it, accelerating toward a block of equal mass which sits at rest. The external force is applied the entire time at the moment of collision and during. One cannot use conservation of momentum or conservation of energy because there's an outside force on the whole system, but I was thinking that during a very short time dt right at the moment of collision you could treat the velocity as constant and the stationary body would obey conservation of momentum over that very short time interval dt, but I can't reason how we represent what happens to the body that is still accelerating and what happens after the interval dt.

I think having been so in the habit of solving these types of problems in a format where the conservation laws hold I am unable to think about how to represent, mathematically, the evolution of the system in a case like this where they do not hold.

Could someone please help elucidate how the equations of motion would be handled in this case?

 

[Orodruin]

The typical thing to do would be to consider the collision short in time. So short that the influence of the external force can be neglected during the collision. If this is a good approximation or not depends on the situation you are trying to describe. If it is not you would typically need to go into more detail about the objects themselves and their properties.

 

[jbriggs444]

If you are not certain whether the collision is of short enough duration that one can neglect the external force then consider calculating the impulse delivered by the external force for the duration of the collision. Multiply (or integrate if the external force will be variable enough to matter) force times duration to compute "impulse".

That is the amount of momentum that is delivered by the external force while the collision is occurring.

 

[MattGeo]

If you are not certain whether the collision is of short enough duration that one can neglect the external force then consider calculating the impulse delivered by the external force for the duration of the collision. Multiply (or integrate if the external force will be variable enough to matter) force times duration to compute "impulse".

That is the amount of momentum that is delivered by the external force while the collision is occurring.

Do you mean by just calculating the impulse starting at the collision instant and incorporating both masses since they're effectively a single mass at that point?

 

[jbriggs444]

Do you mean by just calculating the impulse starting at the collision instant and incorporating both masses since they're effectively a single mass at that point?

If you want an upper bound (highest change of momentum there could possibly be during the collision) then yes, add up the forces on both masses and multiply by the collision duration.