# Calculus 1 : Spatial Calculus

## Example Questions

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### Example Question #167 : How To Find Position

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.

Explanation:

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.

### Example Question #168 : How To Find Position

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.

Explanation:

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!

### Example Question #169 : How To Find Position

Determine the position function of the particle, given the acceleration function of the particle, as well as the following information:

Explanation:

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

### Example Question #170 : How To Find Position

Find the position of the moving particle at t=4 if its velocity function is given by the following:

and its initial position is zero.

Explanation:

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:

### Example Question #1021 : Spatial Calculus

Determine the position of a runner 5 seconds in the future who's accelerating constantly at a rate of , with an initial velocity of  and initial position of

The formula for position given acceleration, initial velocity, and position is:

, where  is position,  is acceleration,  is initial velocity, and  is initial position.

Explanation:

We can describe the acceleration as:

. To find position, we use our formula:

At ,

### Example Question #1022 : Spatial Calculus

If v(t) represents the velocity of a water balloon dropped off a sky scraper, find the function which represents its position if  .

Explanation:

If v(t) represents the velocity of a water balloon dropped off a sky scraper, find the function which represents its position if  .

We are given a velocity function and asked to find a position function. This means we will have to integrate. (Recall that velocity is the first derivative of position)

So, we need to evaluate the following integral:

To integrate a monomial, simply increase its exponent by one, and divide by the new exponent.

Now, we are almost done, but we need to solve for c. We use our initial conditions to do so.

So our position function is:

### Example Question #1023 : Spatial Calculus

If v(t) represents the velocity of a water balloon dropped off a sky scraper, find the balloon's position after 3 seconds, if  .

Explanation:

If v(t) represents the velocity of a water balloon dropped off a sky scraper, find the balloon's position after 3 seconds, if  .

We are given a velocity function and asked to find a position function. This means we will have to integrate. (Recall that velocity is the first derivative of position)

So, we need to evaluate the following integral:

To integrate a monomial, simply increase its exponent by one, and divide by the new exponent.

Now, we need to solve for c. We use our initial conditions to do so.

So our position function is:

Finally, plug in 3 for t to get our answer:

So after 3 seconds, the balloon is still 202.4 units from the ground.

### Example Question #1024 : Spatial Calculus

Given the position function of a projectile, using calculus concepts, at what point in time does the projectile reach its highest altitude? What is this altitude (in meters)?

t=1, 2m

t=1, 3m

t=1.5, 3m

t=0.5, 7m

t=1, 7m

t=1, 7m

Explanation:

To find the highest altitude of a projectile given the position function, we must know where the projectile's direction changes from increasing to decreasing. That is, where the projectile reaches a maximum altitude. This occurs at the point when the velocity is equal to zero.

The velocity of the projectile is modeled by the first derivative of the position function.

The velocity is set equal to zero to solve for t.

The velocity is equal to zero at t=1.

This means that at t=1, the projectile reaches its maximum altitude.

This altitude is solved by solving for s(1).

The highest altitude is 7m and it occurs at t=1.

### Example Question #1025 : Spatial Calculus

Find the position of an object at t=5, given its velocity function  and an initial position of zero.

Explanation:

To find the position function, one must first take the integral with respect to t of the velocity function,

.

Using the rules of integration

where C is some constant and therefore,

.

Since the initial condition of position is given at zero we can solve for C.

Then, by plugging in 2 for t, the correct answer

is obtained.

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