Determine the mass of the first object and the final velocity of the third object on the attached diagram. Regarding Collisions. I've been having...

In any collision, the linear momentum of the system is conserved. By considering the initial (before collision) and final (after collision) momenta of the system, and making them equal to each other, we can find the unknown mass of the first object.

It appears that the third object does not participate in collision, so it's final velocity will remain the same: `vec v_(3f) = vec0` .

The initial momentum of the system consisting of the two objects is

`vecP_i = m_1vecv_(1i) + m_2vecv_(2i)`

Projecting this onto the x-axis (horizontal axis directed to the right) gives

`P_(ix) = m_1(2) + 19*(-5) = 2m_1 - 95`

Similarly, the final momentum of this system is

`vecP_f = m_1vecv_(1f) + m_2vecv_(2f)`

Projecting this to the x-axis gives

`P_(fx) = m_1(-1) + 19(0) = -m_1`

Since the initial momentum and final momentum are equal, `P_(ix) = P_(fx)` :

`2m_1 - 95 = -m_1`

`3m_1 = 95`

From here, `m_1 = 95/3 = 31.7` kg.

The mass of the first object is 31.7 kg.