The package will need to at least gain  in height, thus it will need to gain

 gravitational potential energy

When it is launched, it will have a kinetic energy equal to

Since energy is conserved:

Solving for 

All of the potential energy that the ball has at the top of the ramp is converted into kinetic energy. Knowing this, we can determine the velocity when the ball is moving at the bottom of the ramp by assuming all the potential energy is converted into kinetic energy.

The ball will gain velocity in the vertical direction due to gravity.

The ball will also have velocity due to it's initial horizontal velocity

using

Combining equations and plugging in values:

Neglecting air resistance, we can say that the energy stored in the bungie when it is at its maximum stretched distance is equal to the initial gravitational potential energy of the jumper:

The initial height is the length of the bungie when it is fully stretched. We can write this out as:

 is the length of the unstretched bungie, and  is the distance the bungie stretches. Plugging this into the original expression, we get:

Rearranging for the mass of the jumper, we get:

We have all of our values, allowing us to solve:

First, we calculate the kinetic energy of the block as it slides:

We know that all of the kinetic energy is converted to spring potential energy during the collision. We can use the equation for the potential energy of a spring and set it equal to the kinetic energy. Then, solve for the compression distance:

This is a conservation of energy problem. First, we can set initial energy equal to final energy due to the law of conservation of energy. Therefore,  can be broken into its components:

Because the initial velocity is  and the final height is 0 m, the equation can then be simplified:

From here, we can rearrange the equation to solve for the final velocity:

From here, we can plug in the known values and calculate the final velocity of the roller coaster car:

First, let's set up an axis for this problem. Have the positive direction be towards the outfield and the pitcher and the negative direction be towards home plate and the batter. Next, identify the given information: 

 (This value is negative because the ball is moving towards home plate.)

 (This value is positive because the ball is moving towards the outfield.)

The impulse of the baseball is equal to its change in momentum:

Momentum is equal to mass times velocity, so this equation can be simplified further:

By plugging in the given information, we can solve for the impulse:

Use conservation of energy:

Initially there is no gravitational energy, and in the final state there is no longer spring potential energy.

Solve for the spring constant:

Plugging in values:

The spring force is going to add to the gravitational force to equal zero.

Plugging in values:

Solving for

Using

Plugging in values:

Energy is conserved throughout this entire problem, before the ball is dropped all of the energy is in the form of potential energy represented by , halfway down there is potential and kinetic energy present. Energy at the start must be equal to the energy halfway down which gives rise to the equation:

where  is the initial height and  is the height halfway down. Mass can be cancelled out and the heights and gravity constant can be plugged in in order to find velocity. Plugging in the values: