# Is Impulse Frame Independent?

The title says it all.

I was wondering if the impulse on an object would change if we look from another inertial or non-inertial frame of reference. According to me it shouldn't, since the force that causes the impulse would be frame independent as well. I tried looking it up, but couldn't find it. So can someone prove or disprove the frame-independency of impulse caused by a force?

• What do you mean with "Impulse" ? Momentum ? Momentum is frame dependent. Momentum is a vector which changes under coordinate transformations. Jul 10 '19 at 6:39

In Newtonian physics the impulse is frame independent. In an inertial frame the impulse is:

$$\Delta \mathbf p = \int \mathbf F(t)dt$$

and this does not depend upon velocity so the integral is the same for frames with different velocities. This comes down to the fact that while velocity is frame dependent a velocity difference is not.

For a non-inertial frame we can introduce a fictional force $$\mathbf F_f$$ so the change in momentum measured in this frame becomes:

$$\Delta p = \int (\mathbf F_f(t) + \mathbf F(t))dt$$

But since the forces add linearly this splits into a fictional impulse change and the real impulse change:

$$\Delta p = \int \mathbf F_f(t)dt + \int \mathbf F(t)dt$$

And the second term remains frame independent.

However if we consider relativistic effects then this is no longer true since relativistic speeds do not add linearly and the difference between two velocities is not frame independent. However if we use four-momentum instead and define the four-impulse to be the change in four-momentum then we will find the four-impulse is frame independent.

• So in non-inertial frames of reference (in Newtonian mechanics), the impulse provided by the pseudo/fictitious force may or may not be frame dependent? Jul 10 '19 at 13:31
• @K.T. the fictitious force depends on your chosen frame, so by varying how you choose the frame you can make your function $F_f(t)$ have any form that you want. You could make your non-inertial frame rotate, or accelerate, or both, and have the rotation and accelerating vary with time. All will have different forms for $F_f(t)$ so the integral will be different. Jul 10 '19 at 14:08
• @JohnRennie: I believe the only time the momentum change is frame independent is in a collision - the before and after velocity frames are the same. Also, if you're going represent the force with an explicit time dependence, then you should probably also include the position and the velocity as explicit dependencies since they're the observables which are changing with time - if the force is changing with time. Jul 11 '19 at 2:24