Determining velocity for a pulley/rope system

[Katatatat]

The Question:

Using a simply pulley/rope system, a crewman on an Arctic expedition is trying to lower a 5.65-kg crate to the bottom of a steep ravine of height 23.9 meters. The 60.9-kg crewman is walking along holding the rope, being careful to lower the crate at a constant speed of 1.50 m/s. Unfortunately, when the crate reaches a point 13.3 meters above the ground, the crewman steps on a slick patch of ice and slips. The crate immediately accelerates toward the ground, dragging the hapless crewman across the ice and toward the edge of the cliff.

If we assume the ice is perfectly slick (that is, no friction between the crewman and the ice once he slips and falls down), at what speed will the crate hit the ground? Assume also that the rope is long enough to allow the crate to hit the ground before the crewman slides over the side of the cliff. At what speed will the crewman hit the bottom of the ravine?​

The Relevant Equations:

F = ma
Equations for one-dimensional kinematics​

The Attempt:

I've looked at the forces for the crewman and crate separately with free-body diagrams, applying Newton's Second Law and using the following formulas to solve for acceleration, although I'm not entirely sure that is correct:

T - m₁g = m₁a
m₂g - T = m₂a​

As for the kinematics portion of the question, I assume the crate will follow a simple one-dimensional formula, while the crewman will have a projectile motion.​

 

[Simon Bridge]

Welcome to PF;
The equations are not correct. There are 3 forces on the crewman.
Which direction does the crewman's weight act?

As for the kinematics... don't assume: the problem tells you what sort of motion to expect.