Since we are asked for the shortest home run, the final height of the ball will be the height of the wall. We can then use the following expression:

Where,

We can then use the following expression for time:

Where,

Substituting our expression for time into the first expression, and replacing our component velocities, we get:

Rearranging, we get:

Rearranging again:

We have all of these values, so time to plug and chug:

At the minimum velocity, the ball will reach a distance of 100 meters just when its height is 0 meters.

Let's begin with the following expression to determine how long it takes the ball to hit the ground:

There is no initial vertical velocity, so the expression becomes:

Rearranging:

Plugging in our values:

We can then calculate the initial velocity needed:

Rearranging:

Plugging in our values:

Since the problem statement is presenting a scenario where the initial and final heights are the same, we can simply use the range equation to solve this problem:

Where range is the distance to the home run wall. Rearranging for velocity, we get:

Plugging in our values, we get:

Since we know how far into the stands the ball lands, we can calculate the height that the ball lands at:

Plugging in the horizontal distance, we get:

Then we can use the following expression to determine the initial vertical velocity:

Rearranging for initial vertical velocity, we get:

Plugging in our values, we get:

We can then use this velocity and the initial angle that the ball is hit at to calculate the total initial velocity of the ball (we could also calculate the horizontal velocity of the ball, and then the total initial velocity):

Plugging in our values:

We can use the following piecewise function to determine the height of the ball as a function of the horizontal distance it has traveled:

If , then 

If , then 

We can calculate the final vertical velocity of the ball using the horizontal distance traveled:

Then we can use the following expression to determine the time that the ball is in the air:

Plugging in our values we get:

Simply put:

Using the quadratic formula, we get:

 or 

Thinking practically, the shorter time is when the ball initially hits a height of 14m and the second time is when the ball comes back down to 14m. Therefore,

We can use the following piecewise function to determine the height of the ball as a function of the horizontal distance it has traveled:

If , then 

If , then 

First, we need to determine if the ball will land in the field in the stands or the field. We can use the range equation to get a good idea about this:

Plugging in our values:

From this, it looks as though the ball will be landing in the stands. But wait! There's the home run wall that is 5 meters tall, and from the range equation, the ball would theoretically reach it's original height just shortly after passing the point of the wall. Therefore, let's see if the ball will be making it over the wall. To do this, we will assume a total horizontal distance traveled of 100 meters.

Then we can use the following kinematic expression to determine the final height of the ball when it reaches the wall:

Where:

Where x is the horizontal distance traveled before the ball lands.

Substituting in expressions for our component velocities, we get:

Then substituting time into the kinematic expression:

Plugging in our values, we get:

The ball will hit the home run wall rather than making it over. Therefore, the horizontal distance traveled is 100 meters.

We can use the following piecewise function to determine the height of the ball as a function of the horizontal distance it has traveled:

If , then 

If , then

First, we can guesstimate if the ball will land in the field in the stands or the field using the range equation:

Plugging in our values:

Therefore, the ball will reach its original height at a distance of 90 meters and will land in the field.

There's a few ways to solve this problem. The following method will avoid using the quadratic formula, and is probably a less used method, so it will be good practice.

When the ball reaches a horizontal distance of 90 meters, we know that it will be traveling with the same speed as it began, but with an angle of  below the horizontal rather than above. We can then use the following expression to calculate the vertical velocity of the ball when it hits the field:

If we dictate a sign convention with down being positive, we get:

It's important to keep as many decimal places as you can at this point.

Then, we can use the two velocities to determine the time it takes to travel that vertical distance:

It's a good idea to keep more decimal places than that in your calculator.

Then, we can multiply this time by the horizontal velocity to get the additional horizontal distance traveled:

Plugging in our values:

Then adding our two distances together, we get:

We can use the following piecewise function to determine the height of the ball as a function of the horizontal distance it has traveled:

If , then 

If , then

First, we can guesstimate if the ball will land in the field in the stands or the field using the range equation:

Plugging in our values:

This means that the ball will reach its original height about 58 meters past the home run wall. Therefore, let's operate on the idea that the ball lands in the stands.

We can begin with the following kinematic equation:

Where we can define time as :

Where x is the horizontal distance traveled before the ball lands.

Substituting in expressions for our component velocities, we get:

Then substituting the 2nd expression into the first:

Now we can use the piecewise function above to replace final height or final distance. It will be much simpler to replace final height, so we'll do that:

Expanding the expression a bit:

Then rearranging, we get:

Plugging in values, we get:

Using the quadratic equation, we get:

Obviously our distance will be positive, so we will go with the second one:

First, we can calculate the height that the fan is from the field:

Therefore, we know that the ball will have a vertical drop of 15 meters.

Then we can use the following kinematic expression:

or,

Plugging in our values, we get:

Since time has to be positive:

We can then multiply this by our horizontal velocity to get the horizontal distance traveled:

Plugging in our values:

We can use the following piecewise function to determine the height of the ball as a function of the horizontal distance it has traveled:

If , then   

If , then   

From the problem statement, we know that the end of the stands are  away from home plate. Then using the above function, we can calculate the maximum height of the stands:

Then we can use the following kinematic expression to begin to work toward our answer:

Where:

So:

So:

Plugging this into the kinematic expression, we get:

Rearranging, we get:

Plugging in our values, we get:

Using the quadratic equation, we get:

Thus:

If the above combination of angles seems odd to you, think about what is specified in the problem statement. We are given initial vertical velocity, not initial total velocity. Therefore, as the angle gets lower and lower, the initial total velocity goes up and up. That's why we can have a very low angle (very high total initial velocity), or an angle slightly above 45 which you are typically taught results in the maximum distance of an object (however, in this scenario, the total initial velocity would be much less than with the lower angle).