How can two equal forces applied on the same mass $m$ cause the same displacement in different amounts of time? in my textbook the concept of power is explained through this example:

*

*a man lifts a bucket containing cement to a certain height;

*a freight elevator can lift the same bucket in less time.

It then says that the work in both cases is the same because:

*

*the displacement is identical;

*if the bucket rises at a constant speed, the force directed upward
(by the law of inertia) in both cases is equal (in absolute value) to the
force-weight.

I have two questions:

*

*How can the force directed upward be equal to the force-weight of the bucket if the bucket moves upwards? If the two forces were equal in absolute value, the resulting force would be null and the bucket would stay still.

*If the force applied by the man and the freight elevator is the same, how can these two equal forces applied on the same mass $m$ cause the same displacement in different amounts of time?

 A: 
If the two forces were equal in absolute value, the resulting force
would be null and the bucket would stay still.

The key in understanding this lies in the stipulation that:

if the bucket rises at a constant speed, [...]

Both buckets move at the same velocity, so by Newton's  Second Law the net force acting on them must be zero (null). They would not "stay still", if they were moving uniformly, they will keep moving uniformly (until some net force starts acting on them)
I think your confusion might arise from the understanding that at some point the buckets will have undergone acceleration, which also by Newton's Second Law requires a net force.
The elevator bucket travels faster (more displacement in the same amount of time) because at some time it was accelerated more than the other one. But the accelerations aren't the topic of this textbook passage (which does make it all a bit confusing)
A: *

*If you observe the bucket from an inertial frame (i.e., a reference system whose acceleration is 0), then the velocity $\mathbf{v_0}$ of the bucket is going to be the observed velocity from the inertial frame, $\mathbf{v_{meas}}$ plus the velocity of the frame itself (to make things easy, assume you're standing on the ground and that your velocity has modulus 0, i.e., with speed $||\mathbf{v_f}||=v_f$=0 m/s). Then, since the bucket's speed is constant, you have $$\mathbf{v_0}=const=\mathbf{v_{meas}}+\mathbf{v_f}=\mathbf{v_{meas}}.$$
The problem arises when the force-weight starts acting, making velocities non-constant (a non-null force "adds" an acceleration to the system). But since $\mathbf{v_0}$ is constant, then either the force-weight $\mathbf{F_w}$ is null or its contribution is cancelled by another force $\mathbf{F_{rise}}$ acting to the opposite direction with the same speed:
$$\mathbf{F_w}+\mathbf{F_{rise}}=0\iff \mathbf{F_w}=-\mathbf{F_{rise}}.$$
That's the math, but what's the conclusion? It's all based on $\mathbf{v_0}$, the initial velocity of the bucket. Apparently, it was already moving in such direction before the two forces acted. It doesn't make much sense in a real scenario, but I guess that's it...


*I am not totally sure about this one, but I'm sure that comments and downvotes will do their job properly. Anyways; consider the second law of motion, which includes the sum of all forces acting on a system being equal to the mass times the acceleration of the system. The system composed by the man + the bucket has a certain mass $M_1$, while the system composed by the elevator + the bucket has mass $M_2$. Since it takes less time for the elevator to reach the destination point than it does for the man force, it's safe to say that the accelerations of the two systems are different (if the initial velocity is the same), and since the forces are equal (same work with same displacement=>same force), then necessarily $M_1\neq M_2$.
However, it is not explained how the man lifts the bucket, so we can speak only in numbers: in any case, $M_2<M_1$, since the elevator arrives before and therefore it has a higher acceleration. Since the work is the same, the answer to your question is that the elevator + the bucket consinst of a total mass that is less than that of the system composed by the man lifting the bucket.
Actually, this is not totally correct, because the elevator virtually decrements the bucket's mass due to the normal force exerted by the bucket from the elevator floor, while in the case of the man, both the bucket and the man could exert the normal force in the floor, depending on how the process is done. As such, the actual masses relation may be different, but the "virtual" one still stays $M_2<M_1$.
Edit: I've considered $\frac{d}{dt}\mathbf{v_0}=0$ since the forces are equal and they are not expressed as a function of time. The whole thing changes if the elevator changes speed and therefore undergoes an acceleration, which makes it obvious why it would reach the point faster.
A: 

*

*How can the force directed upward be equal to the force-weight of the bucket if the bucket moves upwards? If the two forces were equal
in absolute value, the resulting force would be null and the bucket
would stay still.


The fact that the net force on an object is zero does not mean the object can't move. It means the object cannot accelerate. It can move at constant velocity. If you drive your car at a constant speed on the road it means the net force acting on your car is zero. (The forces of the drive train pushing you forward equal the opposing dissipative forces of friction (air resistance, kinetic friction, rolling resistance, etc.)) Of course a net force was required to initially get the car moving, as discussed below.



*If the force applied by the man and the freight elevator is the same, how can these two equal forces applied on the same mass $m$
cause the same displacement in different amounts of time?


The forces applied by the man and the elevator are the same only when the bucket is rising at constant velocity. That's because the upward force exerted by the man and the elevator equals the downward force of gravity.
But the fact that the bucket is rising at a greater constant velocity in the elevator is because the net upward force caused by the elevator starting at rest was greater than the net upward force caused by the man starting at rest. That made the net work done by the elevator greater than the net work done by the man until the velocities of each became constant. Per the work energy theorem, the net work done on an object equals its change in kinetic energy.
Once the bucket reached its final velocity by the man and elevator, the upward forces exerted by both were reduced so as to exactly equal the downward force of gravity. From then on, the net forces are zero and the velocities (and kinetic energies) are constant, albeit different accounting for the different elapsed times for the same displacement.

related to the second part you wrote in bold, why do the book assume
that the work performed by the man and the lift are the same? Is that
a simplification or an error?

The work is only the same for the same displacement occurring at constant speed (no change in kinetic energy) as stated in point 2 of the book's reasoning. The work is not the same when the bucket reaches the same height by the man and elevator starting from rest. The gravitational potential energy is the same at the same height no matter how quickly it gets there, but the kinetic energy given the bucket by the elevator is greater than the kinetic energy given the bucket by man because its velocity at the same height is greater. That means the overall net work done by the elevator is greater than the overall net work done by the man.

The book itself uses this example in order to explain the concept of
power saying that the lift exerts a greater power than the man.

The power delivered to the bucket by the elevator is greater than that delivered by the man because the elevator causes the same displacement of the bucket at constant speed as the bucket  but in less time. Power is work per unit time.

The thing that I'm confused about anyway is why does it continue to
assume that their work is the same!

I agree, from what you have presented, that the book may be confusing if it is not differentiating between (2) the total work done by the elevator raising the book from rest to a certain height, which is greater than that done by the man because the bucket has greater kinetic energy at that height, and (2) the work only associated with the same vertical displacement that occurs during constant speed, which is the same for the elevator and the man.
Hope this helps.
