Exploring Collisions w/ Accelerating Objects

In summary: If you want an estimate for the maximum change in momentum that could occur then you would take the integral of the force over the collision duration.
  • #1
MattGeo
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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?
 
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  • #2
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.
 
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  • #3
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.
 
  • #4
jbriggs444 said:
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?
 
  • #5
MattGeo said:
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.
 

FAQ: Exploring Collisions w/ Accelerating Objects

1. What is the purpose of exploring collisions with accelerating objects?

The purpose of exploring collisions with accelerating objects is to understand the principles of motion and energy, as well as how objects interact with each other during collisions.

2. How do you measure the acceleration of an object during a collision?

The acceleration of an object during a collision can be measured by using a device called an accelerometer, which measures the change in velocity of an object over a specific time interval.

3. What factors affect the outcome of a collision between two objects?

The outcome of a collision between two objects is affected by factors such as the mass, velocity, and angle of the objects, as well as the type of material they are made of and the presence of external forces.

4. Can the conservation of momentum be applied to collisions with accelerating objects?

Yes, the conservation of momentum can be applied to collisions with accelerating objects. This principle states that the total momentum of a system remains constant before and after a collision, as long as there are no external forces acting on the system.

5. How does the elasticity of objects affect the outcome of a collision?

The elasticity of objects refers to their ability to deform and return to their original shape after a collision. In a perfectly elastic collision, the objects will bounce off each other with no loss of kinetic energy, while in a perfectly inelastic collision, the objects will stick together and there will be a loss of kinetic energy. The elasticity of objects can greatly affect the outcome of a collision, as it determines how much energy is transferred between the objects during the collision.

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