Determining magnitude of final velocity:

Plugging in values:

Using definition of kinetic energy:

Plugging in values:

Since there was no initial kinetic energy:

Finding total distance:

Using

Plugging in values:

There is no net work done on the ball. The wall stopped the ball, doing negative work, then accelerated the ball, doing positive work. These end up canceling each other out.

Using

Determining initial kinetic energy:

Combining equations

Converting to and plugging in values:

Determining final kinetic energy:

Combining equations

Converting to and plugging in values:

Plugging in values:

We can use the expression for conservation of energy to solve this problem:

Given the problem statement, we know that the ball has no velocity at the initial and final states, so we can remove kinetic energy from the equation:

The initial potential energy is stored in the springs and the final potential energy is gravitational, so we can write:

We multiply by 25 because there are 25 springs.

Rearranging for mass, we get:

We know all of these values, allowing us to solve:

For this question, we're told that an object is normally lifted a certain vertical distance. However, a ramp is to be added in order to make it easier to lift the object to the desired position, and we're asked to find out how much easier it will be by determining how the force required to lift it will change.

First, we must approach this problem from the perspective of energy. Specifically, we need to look at the change in gravitational potential energy. When the object is lifted a certain distance, its gravitational potential energy will increase according to the following expression:

Furthermore, it's important to realize that the change in mechanical potential energy only cares about the final and initial positions; it does NOT care about the path taken to get from initial to final. Therefore, the change in mechanical potential energy for lifting the object directly up is exactly the same as if the object were to be moved up a ramp to the same vertical location.

What's more is that we can realize the gravitational potential energy will be increasing as it is lifted, thus we need to put energy into this process by doing work. We can write the expression for work as follows:

Since the amount of work done on both processes is the same, we can set the two expressions equal to each other as follows:

By rearranging, we obtain:

The expression shown above tells us how the force needed to transfer the object via the ramp is different from the force needed to lift the object directly.

Thus, the force needed to move the object via the ramp is halved compared to the original force. Essentially, since we doubled the distance, we halved the force.

Work is equal to force multiplied by the distance traveled in the direction that the force is applied. The force in this problem is equal to the gravitational force that is created by the weight because that is the minimum force that the weightlifter must apply in order to move the weight. The distance is the distance from the weightlifters chest to the peak of his lift. 

Using

All energy will be kinetic:

The initial velocity will be zero as the ball is motionless

Combining equations:

Using:

Combining equations:

The initial velocity is zero

Combining equations and solving for

Converting to

Plugging in values:

Converting  to 

Converting  to 

Since the initial velocity is zero, initial kinetic energy is zero

Plugging in values:

Plugging in values

Using

Using

Using

Initially the space ship is motionless, so it has  kinetic energy