How can the momentum be conserved in $y$ direction here?

Question

In this question,

A circus acrobat of mass $M$ leaps straight up with initial velocity $v_0$ from a trampoline. As he rises up, he takes a trained monkey of mass $m$ off a perch at a height $h$ above the trampoline. What is the maximum height attained by the pair? [Source: Introduction to Mechanics, D.Kleppner and R.Kolenkow, Chapter 3, Exercise 3.5]

Here, in the solution, the momentum is conserved along the Y-direction by finding out the velocity at height h and then equating the momenta at highest point with it.

However, my doubt is that, momentum can be conserved only when there is no external force acting along that particular direction ; but then here, the weight acts downwards always, so how is momentum conserved?

Please help me understand where I'm going wrong.

Answer 1

The conservation of momentum is used to know the velocity of the pair monkey + acrobat just after they join. The momentum immediatly after the pair is formed must be equal to the momentum just before the acrobat take the monkey.

The rest of the problem is solved by uniformly accelerated movement.

Answer 2

It is true that the momentum of the acrobat and monkey are not conserved, owing to the effect of gravity. However, you can still apply the principle of conservation of momentum in the presence of a force. What you have to do is to account for the effect of the force on the momentum; you can then say that apart from that effect, momentum is conserved.

In the case you mention, gravity acts to change the momentum of the acrobat, so you can work out what speed the acrobat will be moving at when they reach height h. You can then apply to principle of conservation of momentum to say that at the instant the acrobat takes the trained monkey off the perch, the monkey moves at the same speed as the acrobat, therefore some momentum must be transferred from the acrobat to the monkey, and the acrobat must slow down as a consequence of that transfer, an effect that is quite separate from, and in addition to, the continuous reduction in the acrobat's vertical speed that results from the effect of gravity.

To put it another way to stress the point, the acrobat's upward speed is reduced by two effects, one being the continuous gravitational force, the other being a one-off sharing of momentum with the monkey. The former effect is the result of an applied external force; the latter effect is a requirement of conservation of momentum.