# Calculus 1 : Spatial Calculus

## Example Questions

### Example Question #31 : How To Find Velocity

Given the following path:

What will be the velocity  of someone driving along that path?

Explanation:

To find the velocity we need to differentiate each of the components of the position with respect to time.

We note first that:

.

The velocity is then given by:

.

Using the above derivatives, we find the velocity:

### Example Question #32 : How To Find Velocity

A car is moving following the speed law:

What is its acceleration?

No, they will never collide.

Explanation:

To find the expression for the velocity, we need to differentiate each of the components with respect to time.

We have:

Collecting the components we obtain the expression for the velocity:

### Example Question #33 : How To Find Velocity

The velocity of a particle is given by:

When is the above expression defined?

Explanation:

To have V defined for all t, we need to have each of the components defined as well.

First we see that for the first component to be defined , we need to have .

For the second component we need to have .

The last component must satisfy .

This means that t must satisfy all these 3 restrictions.

This gives :

This means that

### Example Question #34 : How To Find Velocity

The position of a moving ant is given by:

Find its speed.

Explanation:

To find the speed, we need to find the velocity first. We are given the position vector. We need to differentiate to get the velocity vector and find its norm to find the speed.

To find the velocity we differentiate componentwise.

Using the Chain Rule we find the derivative to be:

We take the norm of the above vector to get :

### Example Question #35 : How To Find Velocity

A car is moving through the following path:

Find its speed at any given time.

Explanation:

To find the speed, we need to find the velocity first. We need to differentiate the expression of the position vector componentwise with respect to time.

Knowing that .

We have the velocity given by:

.

To find the speed we will have to find the norm of .

and finally this gives:

### Example Question #36 : How To Find Velocity

Given the vector position:

Find the expression of the velocity.

Explanation:

To find the expression of the velocity vector, we must differentiate each of the position vector components  with respect to time.

We have then:

Finally collecting all these derivatives componentwise, we obtain the velocity.

### Example Question #37 : How To Find Velocity

Given the position, , find the velocity at time .

Explanation:

To find the velocity given the position function, take the derivative of the position function using the Power Rule.

Applying the Power Rule to our function we get the following.

Substitute  to determine the velocity at this time.

### Example Question #38 : How To Find Velocity

Suppose the position function of an object is .  What is the velocity of the object when

Explanation:

To find the velocity of , take the derivative of this function to find .

Recall that natural log has a special derivative.

Substitute  and find the velocity at this time.

### Example Question #39 : How To Find Velocity

The position of an object is described by the function .  What is the average velocity of this object from  to ?

Explanation:

Write the formula for average velocity.

The variables  is the final position,  is the initial position,  is the final time, and  is the initial time.

Plug in  and  into  to solve for initial and final position.

Substitute the known variables into the average velocity formula.

### Example Question #40 : How To Find Velocity

Find the velocity function of a particle if the acceleration function is .