# Calculus 1 : How to find acceleration

## Example Questions

### Example Question #481 : Calculus

For this function, keep in mind that acceleration is measured as .

A particle's position is given by the function . Find the particle's acceleration when .

Explanation:

Acceleration can be found by taking the second derivative of the position function.

The given position function is , its first derivative using the power rule is

, a constant.

Recall that the derivative of a constant always is zero we see that the second derivative of this function is .

Therefore, we have that the acceleration is always .

### Example Question #482 : Calculus

An atom's position is defined by the equation . What is the acceleration of the atom at ?

Explanation:

By definition, acceleration  is the second derivative of a position function .

Therefore, .

For this particular problem we will use the power rule to find the derivative. The power rule states,

Since , applying the power rule once we get  and applying it a second time we find  . Plugging in  gets us .

### Example Question #483 : Calculus

A car's position is defined by the equation . What is its acceleration at ?

Explanation:

By definition, acceleration  is the second derivative of a position function .

Therefore, .

For this particular problem we will use the power rule to find the derivative. The power rule states,

Since , applying the power rule once will get us

Applying the power rule a second time we find .

Plugging in  gets us .

### Example Question #484 : Calculus

A cat's position is defined by the equation . What is its acceleration at ?

Explanation:

By definition, acceleration  is the second derivative of a position function .

Therefore, .

For this particular problem we will use the power rule to find the derivative.

The power rule states,

Since , applying the power rule we get

Thus applying the power rule again we find  .

Plugging in  gets us .

### Example Question #485 : Calculus

What is the acceleration of an object with a position function  at ?

Explanation:

Acceleration is the second derivative of position, or .

Given,

we can use the power rule which states,

.

Applying this rule we get .

From here we can use the power rule again and deduce that

.

Swapping in , we get

.

### Example Question #486 : Calculus

What is the acceleration of a bird with a position function  at ?

Explanation:

Acceleration is the second derivative of position, or .

Given,

we can use the power rule which states,

.

Therefore,

From here we apply the power rule a second time and deduce that,

.

Swapping in , we get

.

### Example Question #487 : Calculus

What is the acceleration of a boat with a position function  at ?

Explanation:

Acceleration is the second derivative of position, or .

Given,

we can use the power rule which states,

.

Applying the power rule we find,

.

From here we apply the power rule a second time and deduce that,

.

Swapping in , we get

.

### Example Question #488 : Calculus

Satoshi is pacing back and forth in an erratic way. His position at any given time is defined by the following function:

Find a function describing his acceleration at any point in time.

Explanation:

We are given the position function of Satoshi; we can find his acceleration function by taking the second derivative of his position with respect to time.

The first step will be to find the velocity function by taking the first derivative of the position function.

The position function is:

To take the derivative of this function, let's handle each term separately.

The derivative of

is

To find the derivative of the

term, remember to find the derivative of the term inside of the parenthesis, and multiply it by the derivative of the outside term:

In this manner we find our velocity function:

To find acceleration, we then take the derivative of velocity with respect to time.

Since the second term, 2tsin(t2), is a product, we must use the product rule for derivatives:

Derivative of ):

Adding this to the derivative of

which is

we can define our acceleration function as:

### Example Question #489 : Calculus

The position of a very erratic particle can be described with the following equation:

Write an equation that describes its acceleration at any point in time.

Explanation:

Acceleration is defined as the second derivative of the position function with respect to time. Taking the first derivative will give us a velocity (remember that for trigonometric functions and exponentials, take the derivative of the inside terms as well and multiply them with the derivative of the outside terms):

Taking the derivative of this, remembering to use the product rule for derivatives, in turn gives us the acceleration:

### Example Question #91 : How To Find Acceleration

The position of a given chicken is defined by the equation . What is the acceleration of the chicken at ?

Explanation:

Acceleration  is defined as the second derivative of position , or

Therefore, given

and using the power rule

where ,

and

Consequently, at

.