The change in position from time 0 to time a is given by

.

By the information given to us in the question for the velocity function and the change in position, we can form the equation:

From here, we integrate the left side:

This equation can be solved for a using the quadratic equation:

Since time must be positive, the answer is 3.

To begin with, start by finding the velocity function, the integral of acceleration with respect to time:

, the integration of constant, may be found by knowing an initial condition. We are told the car starts at rest, meaning:

The distance the car travels in a given time is then given by an integral bounded by the start and finish times:

In this problem, we are given a distance and a starting time, so we must solve for the final time:

We are given that the trains deccelaration equation is  and that its initial velocity is .  With this information we can find the minimum distance the train needs to start breaking in order to avoid hitting the car.  We must realize that the integral of acceleration is velocity and the integral of velocity is position.  

Therefore we must take the integral of the acceleration formula in order to find the train's velocity equation.  

Using the general integral formula 

,

we find that the velocity formula for the train is .

Plugging in  to solve for , we find that , therefore the final velocity equation is .  

Now in order to find the position function, we must once again take the integral of the velocity equation to find the position equation.  

Using the general integral formula 

, we find that the position formula for the train is .  

Plugging in the 5 seconds the driver has to break, we find that the train must be at least 375 meters away in order to avoid hitting the car.

The distance an object has traveled after a given amount of time is the area under the speed curve. To find the area under s(t) between 0 and 1, the following definite integral must be evaluated:

Use u-subtitution to evaluate this integral. Let u = 2t. Then du = 2dt, and dt= du/2. We also need to find the new limits of integration. When t = 0, u = 2(0) = 0. When t = 1, u = 2(1) = 2. Using this information, we can rewrite the integral as: 

Since the antiderivative is:

Evaluating this antiderivative at the limits of integration leads to the final answer:

To find total distance, integrate the absolute value of the velocity function over the specified interval. The easiest way to do this is to break the integral into two separate pieces, one for the region where sin(t) is positive and one for the region where sin(t) is negative.

Recall that sin(t) is positive on the interval from 0 to . The distance travelled over this interval is:

Next, compute the integral of sin(t) over the interval . Note that sin(t) is negative on this interval, so the value of the integral will be negative. After evaluating the integral, take the absolute value of the answer to find the distance travelled. 

 , so the distance traveled is 1.

Putting the distances found from the two integrals together, we find that the total distance traveled is 2+1 = 3.

To compute the change of distance using the velocity equation, you need to take the definite integral of the velocity function.  This would be:

This will require you to evaluate the following function:

, Thus you have:

Remember that the velcoity function can be found by taking the integral of the acceleration function.  Likewise, the position can be found by taking the integral of the velocity function.  We know:

Therefore, we know:

However,  must be , given the initial velocity of the particle.  This means that the distance can be computed:

This means you need to evaluate  at both  and :

The position function for a physical body can be computed using the integral of the velocity function:

Thus, we know that 

.  

Since, at , the position is , we know that .  

Therefore, we can solve for :

Based on our data, we know that the equation for the acceleration of the ship is:

Now, we know that we can find the velocity of the ship by using an integral so that:

This means:

However, since we know that at time , the velocity was ,  must equal .  Therefore, we know:

Now, the distance traveled by the ship during this period can be found by taking the definite integral from  to  of :

Therefore, we need to evaluate the following function for  and :

In order to solve this problem, you must know that the derivative of a position function is the veloctiy function and the derivative fo the velocity function is the acceleration function.  Thefore, by taking the double integral of the acceleration function we will be able to find the position function.  

In order to take the integral of the acceleration function, , we must first know the general rule of integration .  

Applying the general rule of integration, we find the velocity function to be .  Plugging in the known value of the intial velocity of the cannonball, we find that .  We also find that the moment in time that the velocity of the cannonball is equal to 0 is at 6 seconds.  This is the moment in time the cannonball stops moving, we will need to plug this into the position function to find how far the cannonball traveled.

Taking the integral of the veloctiy function with the general rule of integration, we find that position function becomes .  Since the intial position of the cannonball is 0, the position function becomes .  

Now by plugging in 6 seconds, we can find how far the ball traveled, which in this case is .