# Constant momentum of an accelerated body

1. Apr 3, 2014

### Michael_25

Momentum of a body can be constant while it accelerates? I mean if velocity increases while mass decreases proportional.
And if is true, what force produces acceleration?

2. Apr 3, 2014

### Staff: Mentor

That is an interesting idea.

$$p=mv$$
$$\frac{dp}{dt}=m\frac{dv}{dt}+v\frac{dm}{dt}$$

So to keep momentum constant would require mass to change as:
$$\frac{dm}{dt}=-\frac{ma}{v}$$

3. Apr 3, 2014

### Michael_25

Yes, but what produces that acceleration?

4. Apr 3, 2014

### Staff: Mentor

Any force would be fine. Have you studied Newton's 2nd law: $\Sigma f = m a$

5. Apr 3, 2014

### D H

Staff Emeritus
An external force could cause it, but an external force is not needed. That acceleration could come from a variable specific impulse rocket. Consider a rocket in deep space, far removed from any external forces. The rate at which the rocket's momentum changes is $\dot p = \dot m (v-v_e)$. So all we need to do to keep the momentum constant is to keep increasing the exhaust velocity $v_e$ in tune with the rocket's velocity $v$.

6. Apr 3, 2014

### A.T.

This is kind of like shooting backwards while moving, so the projectile has zero final momentum. Therefore the empty cannon cannot have lost/gained any momentum compared to the loaded cannon. The cannon's gain in velocity is canceled by the loss of the mass of the projectile.

Last edited: Apr 4, 2014
7. Apr 3, 2014

### Michael_25

In my opinion is a misconception that the initial momentum of a body can be constant while it accelerates.
Let be initial momentun of the body $p_1=m_1v_1$. The body splites in two bodies, with the momentum $p_2$ and $p_3$, so that $p_2+p_3=p_1$.
If we put the condition $p_1=p_2$ = constant, then we got $p_3=0$.
But we know to produce an acceleration we need a force $F=\frac{dp}{dt}$. If $dp=0$ (like $p_1=p_2$), there is no force and no acceleration.

8. Apr 3, 2014

### Staff: Mentor

It would be very unusual, but certainly not impossible provided the conditions above are met.

Which is exactly what D H said in post 5 and A.T. said in post 6.

This equation is only correct if $\dot{m}=0$

9. Apr 5, 2014

### Michael_25

I think it is required to establish that the claim "momentum of a body can be constant while it accelerates" is valid for a system of bodies and not for the same body, because initial body splits in other bodies, as it accelerates.