# Calculus 2 : Applications in Physics

## Example Questions

### Example Question #201 : Integrals

Suppose a particle has an acceleration given as the following function of time: . At time , the velocity of the particle is .

After one second, how far has the particle moved from its original position?

Explanation:

In order to figure out the particle's position at time , we must first discover the position as a function of time.

We know that

,

and that the velocity at time is given by

.

Since the initial position is not known, we might as well say that the initial position is at the origin, that is to say that

.

Then will tell us how far the particle has traveled from the origin, which would be exactly the distance the particle has traveled from it's original position. First, we must compute . We know that,

so then

,

where is some constant. To find , we use the condition that . Plugging this in to the velocity function, we get

.

Thus our velocity as a function of time is

.

Now to find position, we know that

.

Plugging in , we get

,

where again is some unknown constant. To solve for , we plug in the condition and obtain .

Thus our position as a function of time is given by .

Now it is just a matter of plugging in . Doing so, we obtain .

Therefore our distance away from where the particle started is .

### Example Question #202 : Integrals

A particle starts at rest at position zero, and its velocity is described by  What is the position of the particle as a function of time?

Explanation:

The position of a particle can be found by integrating the velocity function.  We need to use the initial condition to find our constant, since it will be an indefinite integral.

Therefore, our constant equals

Putting it all together, our equation is:

### Example Question #203 : Integrals

If and , what is the original position function?

Explanation:

First, integrate the velocity function because the integral will be the position function: . Then, integrate. Remember to raise the exponent by 1 and then put that result on the denominator: . Then, plug in your initial conditions to find your C: . Simplify and solve for C. . Plug that in to get your position function: .

### Example Question #204 : Integrals

If and , what is the original position function?

Explanation:

First, write an integral expression. Remember that the integral of velocity is position:

Then, integrate. Remember to raise the exponent by 1 and then also put that result on the denominator. Also, add a C because it is an indefinite integral at this point:

To find C, plug in your initial conditions:

Solve for C:

Plug in to get your position function:

### Example Question #205 : Integrals

If and , what is the original position function?

Explanation:

First, write an integral expression. Remember that the integral of the velocity function is the position function:

Integrate. Remember to add one to the exponent and then also put that result on the denominator. Also, remember to add C because it is an indefinite integral at this point:

Now, plug in your initial conditions to find C:

Now, solve for C:

Sub C into your position function:

### Example Question #211 : Integrals

If and , what is the original position function?

Explanation:

First, integrate the velocity function to get the indefinite position function:

Now, plug in your initial conditions to solve for C:

Now, plug back into the position function:

.

### Example Question #212 : Integrals

If and , what is the original position function?

Explanation:

First, write the integral expression for this problem:

Now, integrate. Remember to add one to the exponent and then put that result on the denominator:

Now, plug in your initial conditions to find C:

Now, plug your C into the position function:

.

### Example Question #213 : Integrals

If and , what is the original position function?

Explanation:

First, set up the integral expression:

Now, integrate. Remember to raise the exponent by 1 and put that result on the denominator. Also remember to add a C because it is an indefinite integral:

Plug in your initial conditions to solve for C:

Plug back in for C and get your position function:

.

### Example Question #214 : Integrals

If and , what is the original position function?

Explanation:

Remember that the integral of the velocity function is the position function. First, write the integral expression:

Integrate. Remember to raise the exponent by 1 and then put that result on the denominator:

Now, plug in your initial conditions to solve for C:

Plug back in to get your position function:

### Example Question #215 : Integrals

If and , what is the original position function?