If the particle is said to have returned to its starting point, that would mean that the total distance traveled is zero.

Distance traveled over an interval of time can be found by integrating the velocity function over that interval:

For the velocity function

Setting this equation equal to zero will provide the times that the particle returns to its starting point:

The equation is equal to zero at times , though we can disregard the trivial solution of .

The particle returns to its starting point for the first time at time

Distance traveled over an interval of time can be found by integrating the velocity function over this interval of time:

For the velocity function

A distance formula can be found as

This function is equal to zero at times  Discounting the trivial solution of , the particle first returns to its starting point at time

Distance traveled over an interval of time  can be found by integrating the velocity function over this interval:

For velocity function

The distance over the interval of time  is

The velocity function can be found by integrating the acceleration function with respect to time. For the acceleration function

The velocity function is

The constant of integration can be found by using the initial condition given in the problem, that is :

Now, average velocity can be found by dividing distance traveled by total elapsed time. Furthermore, distance can be found over an interval of time by integrating the velocity function over said interval. Combining these two facts, average velocity can be given as:

In the case of this problem:

Note that the starting position was irrelevant.

Distance traveled over an interval of time  can be found by integrating the velocity function over this interval:

For the velocity function

Over the interval of time , the distance traveled is

Distance traveled over an interval of time  can be found by integrating the velocity function over this interval with respect to time:

For the velocity function

The integral of a constant  is simpy , and the integral of an exponential function  is .

We'll treat the ending time as unknown and write a general distance equation:

The time when he crosses the finish line is when this equation is equal to :

This occurs at time

Begin by finding the velocity function, which is the integral of the acceleration function with respect to time.

For acceleration

The integral of a constant  is simpy , and the integral of an exponential function  is .

The velocity function can be found using these properties to be

The constant of integration can be found utilizing the initial condition:

This gives the complete function

Now to find distance:

Distance traveled over an interval of time  can be found by integrating the velocity function over this interval:

Over the inteval 

Note that the integral of a function  is . 

Distance traveled over an interval of time  can be found by integrating velocity over this interval:

For the velocity function

We do not know the end times, but we can assume that the timer starts at zero. Therefore, we can write a distance equation of the form:

The integral of a constant  is simpy , and the integral of a function of the form  is ., where 

Now, begin by finding the time it takes Lacey to run fourteen feet:

So there are two roots ; however, a negative time is not possible, so she runs the first fourteen feet in two seconds.

Now, calculate the time it takes her to run twenty-eight feet:

The positive root is 

So the time it takes Lacey to run the additional fourteen feet after the first fourteen feet is the difference of these two times:

From the position equation 

we see that time is in the unit of seconds.  Therefore, we must convert  into .  Since there are  in  and  in , the total amount of time in seconds in .  Plugging into the position equation gives us .  After rounding the answer, we get  or 

.  

From the passage, the position the ball travels across the Mississippi River is

From this equation, we let  and plug into the equation to get the distance traveled ()