In order to determine the position function from the velocity function, we need to integrate the velocity function.

Integrate the velocity function.

Since the initial condition is zero, substitution of  and  will leave , which will eliminate the constant term.  Rewrite the position function.

Solve for .

To solve, integrate to find position and then plug in . Thus,

To find the maximum height of the ball we will first need to find the moment which it has stoped moving. This is the same as finding when the velocity is equal to zero. The velocity at the maximum height will be , so we need to determine when that happens. 

For this particular function we will need to apply the power rule which states, .

Applying the power rule to our position function we are able to find the velocity function.

From here, set the velocity function equal to zero and solve for time.

Now we can find the height of the ball at time .

The original problem tells us that the calculations are in inches, so we need to convert to feet.

.

We are given very little information, so let us determine what we know. 

Note that both of these values are negative because the object is traveling downward. (This is personal preference. You can use down as the positive direction in this problem as well so long as the signs of the velocity and acceleration are the same.)

We know that we can integrate acceleration to derive velocity, so we can find .

Recall the rules for integration that apply to this specific problem,

applying this rule to the accelaration we are able to find the velocity function.

We can now use the initial value for  to determine .

.

Plugging in , we find

.

Again, we can integrate the velocity function to find the position function.

.

We know that , so we can use that to find .

Now we simply need to determine  when the height of the bucket is .

Using the quadratic formula,

where 

we find that 

Because negative time does not make sense, our solution is .

Let us first determine what we are given.

, 

Let's plug what we know into the general formula.

The derivative of position will give us the velocity function. For this particular function we will need to use the power rule to find the derivative.

The power rule states, .

Applying the power rule to the position function we are able to find the following velocity function.

When the arrow reaches its maximum height, the velocity will be . Therefore, set the velocity function that was found equal to zero and solve for time.

 seconds.

Now we can find the position of the arrow at  seconds.

.

To solve, simply integrate the following function and plug in .

To determine the position function, we must integrate the velocity function:

The integral was found using the following rule:

Now, to solve for C, use the initial condition that the position was 4 at t=0. We find that c=4. 

Finally, rewrite the answer, replacing C with the known value:

The following knowns are given in the problem: the velocity of the robber, the initial velocity of the robber and the robot, and the acceleration of the robot. Using these, the expressions for the postions of the robot and the robber can be determined and used to find the position in which the robot catches up to the robber. 

In order to find the position of the robber, the velocity of the robber must be indefinately integrated (in order to find the anti-derivative), since postion is the integral of the velocity. 

Since the bank is the starting point (x=0), the C value can be determined to be zero:

.

Therefore the position of the robber at any given time is given by 10t. This makes sense, because he is moving 10 meters every second. 

The position of the robot is slightly tricker due to the introduction of initial conditions. The robot does not start at x = 0 OR v = 0, therefore the C constants must be determined at each integration. The integral of the acceleration will give the velocity, the intergal of the velocity will give the expression for the robot's position.

The C value is equal to the initial condition of the velocity, meaning the the value of C is equal to 5 m/s. To make it more clear, at t = 0, the velocity of the object has not yet had time to accelerate, meaning the velocity of the object is equal to the velocity it started at. Mathematically, you can set the expression equal to the initial condition and set t equal to zero. So 10(0) + C = 5 gives the same answer of C = 5. Now the postion can be determined using the velocity. 

This expression gives you the postion of the robot at a given time. The constant C is again equal to the initial condition, in this case the initial postion, which is 30 meters. The postion is therefore equal to 5t²+5t+30.

Since the expression for postion is known for both the robber and the robot, they can be set equal to one another, and solved for t to find the time it takes for them to catch up to one another. The value of t can then be used to find the position in which the robot caught up to the robber. Keep in mind that the robber had a minute head start, so his starting postion will be different than the starting postion of the robot. Since he has 60 seconds to move before the robot, he was already at p(60) = 10(60) = 600 meters!

Then using the quadrative formula:

Since the time cannot be negative, this means that 11.19 seconds after the robot is released, he will catch up to the robber! The postion at which the robber is caught can finally be solved to be at:

With three sig figs, position = 

To find , you have to first find the velocity function. That means taking the derivative of the position function. To find the derivative of a term using the power rule, you multiply by the by the power of the exponent and then subtract one from the exponent. becomes , becomes and the derivative of  is  because it is a constant. Therefore, the velocity function is . Then, plug in  to get the value of the velocity function at . .

To tackle this problem, you must integrate the velocity to find the position function. Remember, when integrating, raise the exponent by  and then put that result on the denominator of the  term. Therefore, when integrated, you get: .

Simplify to get . Plug in your initial condition () to find your C: .

Plug your value in for C so that your final answer is: .