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I did this experiment in my lab. We had two cars with wheel connected by a string. One car has a spring that can be triggered by a switch at the top of the car. (Frictionless surface and light string$

We press it and the two cars explode and we measure of the distance travelled by each car and the mass of each.

We then figured out from the ratio of each car's mass-distance product, that the momentum is conservation because the ratio is always 1. (Assuming time for each car to stop is same)

But for the last trial, we flipped a car to creat friction (no wheel). Then the ratio is zero anymore, meaning no conservation of momentum.

My teacher told me to find the force of friction acting on the car , I need to find the difference in the momentum of the two cars and use the time the car with friction takes to stop figure out the force. Why does this way of finding out the force works?

Did frictional force acting on each cart create the same impulse in each cart? Why/not ?

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    $\begingroup$ Neither energy conservation nor momentum conservation fail, your calculation simply doesn't include the energy that is converted into heat and the momentum that is being transferred to the table. $\endgroup$
    – CuriousOne
    Commented Jan 13, 2016 at 0:25
  • $\begingroup$ @CuriousOne good cache its momentum not energy $\endgroup$ Commented Jan 13, 2016 at 0:26
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    $\begingroup$ Ah, you mean "Good catch"? OK, glad we could clarify the question. Momentum conservation doesn't fail, either. The friction force will transfer the momentum to the surface, which transfers it to the table and from there it becomes transferred to the motion of the Earth. This happens to both cars and since the two cars were at rest relative to Earth all of this will, after the cars have come to a rest, all work out to zero momentum, again. $\endgroup$
    – CuriousOne
    Commented Jan 13, 2016 at 0:46

2 Answers 2

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Force is the rate of change of momentum, $$ \vec F = \frac{d\vec p}{dt} = m\frac{d\vec v}{dt} = m\vec a. $$ If you have an external force (like friction), you are exchanging momentum with the outside world. In this case, conservation of momentum demands that the momentum in your system change. Only with zero net external force can you observe conservation of momentum in a system.

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The reason for the law of conservation of momentum is due to the translational symmetry of the Lagrangian in the absence of dissipative forces like friction. $$\frac{d}{dt}\bigg(\frac{\partial L}{\partial \dot{q_i}}\bigg) - \frac{\partial L}{\partial {q_i}} = 0$$ Where translational symmetry and conservation of momentum is implied by $$\frac{\partial L}{\partial {q_i}} = 0 = \dot{p_i}$$

If there is a dissipative force, then the Rayleigh dissipation function is added to the Lagrangian as follows. $$\frac{d}{dt}\bigg(\frac{\partial L}{\partial \dot{q_i}}\bigg) - \frac{\partial L}{\partial {q_i}} +\frac{\partial R}{\partial \dot{q_i}} = D^*_i $$ Where $R$ is the Rayleigh dissipation function, $D^*_i$ is the part of the force $D$ not derivable from a potential energy function, and $L$ is the lagrangian. Obviously an equation of those form will not have any translational symmetry because of the $D^*_i$ term on the right which is due to a non-conservative dissipative force like friction.

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