How do you find the final velocity of two objects coupled?

Answer 1

The case you're describing is called an inelastic collision.

Two objects collide, stick to each other and continue their motion as one body.

Due to momentum conservation principle, sum of two bodies momenta before collision has to be equal to momentum of the one body after collision.

pbefore = pfirst + psecond = m1v1 + m2v2

pafter = (m1 + m2)vcommon

Since pbefore = pafter,

(m1 + m2)vcommon = m1v1 + m2v2

We can get vcommon from that:

vcommon = (m1v1 + m2v2) / (m1 + m2)

[vi are velocities of bodies before collision and vcommon is a velocity after collision]

Answer 2

To find the final velocity of two objects that are coupled, you can use the principle of conservation of momentum. The total momentum before the coupling is equal to the total momentum after the coupling. Set up an equation where the initial momentum of object 1 plus the initial momentum of object 2 is equal to the final momentum of the coupled objects, and solve for the final velocity.