To find the position of the particle at time t=1, we must first find the position function of the particle, which is given by the integral of the velocity function:

The integration was performed using the following rule:

Now, plug in the initial condition to solve for C:

Finally, evaluate the position function at t=1,

For any velocity function, v(t), the position function (r(t) or sometimes p(t)) may be found by taking the indefinite integral of v(t).

Now given the intitial condition , we may solve for our constant "C".

We now have the equation

Evaluating for 

To find the position of the particle, you need to integrate the veloctiy function.  Then, plug in   means the -component of the position, and  refers to the -component of the position, so in the end, we will write the solution in coordinate form

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

The integral was found using the following rules:

, 

Now, to solve for C, we must use the initial condition:

Replacing C with the value, we get

To find the position function from the acceleration function, integrate  twice.

When integrating, remember to increase the exponent of the variable by one and then divide the term by the new exponent. Do this for each term.

Solve for .

In order to find the position function from the velocity function we need to find the anti-derivative of the velocity function 

When taking the integral we use the inverse power rule which states

Applying this rule we get

To find the value of the constant  we use the initial condition

which yields

Therefore

Your roommate is angry at you and decides to throw your phone out the window. v(t) models the velocity of your phone as it falls to the ground. Find the function which models your phone's position.

We are given a velocity function and asked to find a position function. To do so, we need to integrate v(t). (Recall that velocity is the first derivative of position, so to get back to position we need to integrate.)

So, we need to evaluate the following indefinite integral:

Integrating a polynomial is as easy as taking each term, increasing the sponent by 1, and diving by the new exponent. Don't forget the constant of integration as well.

So our answer becomes:

Your roommate is angry at you and decides to throw your phone out the window. v(t) models the velocity of your phone as it falls to the ground. Find the position of your phone after 5 seconds, if the constant of integration is 15.

We are given a velocity function and asked to find a position function. To do so, we need to integrate v(t). (Recall that velocity is the first derivative of position, so to get back to position we need to integrate.)

So, we need to evaluate the following indefinite integral:

Integrating a polynomial is as easy as taking each term, increasing the sponent by 1, and diving by the new exponent. Don't forget the constant of integration as well.

So our funciton becomes:

But we are told that our constant of integration is 15, so add that in:

Finally, we need P(5), so let's find it out!

So our answer is 740 units. Must be a tall dorm building!

To determine the position function, we need the velocity function, which is found by integrating the acceleration function:

The integration was performed using the following rule:

Now to solve for C, we must use the information given for the velocity at t=0:

The velocity function then becomes

To find the position function, we integrate the velocity function:

The integration was performed using the same rule as before.

To find C, we must use the information given for the position at t=0:

The position function is now

To determine the position of the moving particle, we must integrate the velocity function to get the position function:

The integration was performed using the following rule:

Next, to solve for C, we use our initial condition information:

Rewriting the position function, we get

Finally, plug in  to solve for position: