Does a conservation of momentum problem need to specify velocity, not just speed?

Question

Problem: A $100$ g bullet is fired from a $10$ kg gun with a speed of $1000$ m/s. What is the speed of the recoil of the gun?

This problem can be solved, if it can, using the law of conservation of momentum as:

We know that $$m_{gun}u_{gun}+m_{bullet}u_{bullet}=m_{gun}v_{gun}+m_{bullet}v_{bullet}$$

Since $u_{gun} = u_{bullet} = 0$ (Velocities before the shot), therefore

$$m_{gun}v_{gun}+m_{bullet}v_{bullet}=0,$$ $$\mbox{or}\ v_{gun}=-\frac{m_{bullet}}{m_{gun}}v_{bullet}.$$

This shows that the velocity of recoil of the gun is opposite to the direction of the velocity of the bullet. Anyway, to solve this problem I need velocity, and not speed. If I plug in $1000$ m/s for $v_{bullet}$, the formula would think that the velocity of the bullet is in some set positive direction. Even if it is in some set positive direction, the problem needs to state that; it should say velocity, not speed of the bullet. Am I right?

Answer 1

1)The conservation law of linear momentum does involve the velocity, not just the speed. That said, however, it is ok to use speed in the problem statement.

There are no external forces acting on the system bullet-gun. Therefore, if the bullet moves along the positive $x$-axes the gun recoils in opposite sense along the same direction. This is implicitly assumed, or at least it is assumed that you know it/will figure that out. Hence, there is no need to be more pedantic than necessary and talking about speed generates no confusion at all. "You know where things are moving" is the kind of attitude of the problem statement.

In a more general case, it is left for you use the conservation law correctly, without assuming things that aren't true. You did well in this example.

2) It is totally meaningful. In the one dimensional case, $|A|$ corresponds to the absolute value of its single component. There is thus a distinction worth being made.

Answer 2

No, you're not right.

Anyway, to solve this problem I need velocity, and not speed.

Why exactly do you need velocity, considering that the problem asks for the speed of the gun?

In any case, unless there is a predetermined coordinate system you're supposed to use, it's fine to specify a velocity as something like "$1\ \mathrm{m/s}$ opposite the bullet's velocity". Or you can define your own coordinate system, in which case you're free to choose the direction of the bullet's velocity to be positive or negative as you like.