Step 1: Understand apparent weight.
A man of mass 50 kg stands in a lift moving downward and speeding up at $2.8$ m/s$^2$. The floor pushes up on him with a normal force, which we must find.
Step 2: Identify the two forces on the man.
Gravity pulls him down with weight $mg$. The floor pushes him up with normal force $N$. Since the lift accelerates downward, the net force points downward.
Step 3: Write Newton's second law.
Taking down as positive, the net downward force is $mg - N$, and this equals mass times acceleration: $mg - N = m a$.
Step 4: Rearrange for the floor force.
Solving gives $N = m(g - a)$. The floor pushes with less than full weight because the lift drops.
Step 5: Put in the numbers.
With $m = 50$ kg, $g = 9.8$ m/s$^2$, and $a = 2.8$ m/s$^2$: $N = 50 \times (9.8 - 2.8)$.
Step 6: Compute the answer.
$N = 50 \times 7 = 350$ N. \[ \boxed{350 \ \text{N}} \]