# Calculus 1 : How to find acceleration

## Example Questions

### Example Question #11 : How To Find Acceleration

A car is moving at a constant speed of  miles per hour.  What is the acceleration after  hours?

Explanation:

The car is moving at a constant speed of 40 miles per hour, so the velocity function is:

The derivative of the velocity function is the acceleration function.

The acceleration at any particular time is zero.

### Example Question #11 : How To Find Acceleration

The position of an object, in meters, is given by the following equation:

Find the acceleration of the object.

Explanation:

Velocity is the derivative of position, and acceleration is the derivative of velocity, so acceleration is the second derivative of position. With that in mind, all we have to do to find the acceleration of the object is take the derivative of the equation for its position twice.

### Example Question #13 : How To Find Acceleration

The velocity of an object is given by the following equation:

Find the equation for the acceleration of the object.

Explanation:

Acceleration is the derivative of velocity, so in order to find the equation for the object's acceleration, we must take the derivative of the equation for its velocity:

We will use the power rule to find the derivative which states:

### Example Question #14 : How To Find Acceleration

The acceleration of an object is given by the folowing indefinite integral:

If , find the acceleration of the object at  seconds.

Explanation:

In order to find a(3), our first step is to evaluate the integral in the equation for acceleration:

Now we use the initial acceleration, a(0)=0.1, to solve for the constant C:

So if C=0.1, then our final equation for acceleration is as follows, which we can then plug t=3 into to find the acceleration of the object after 3 seconds, a(3):

### Example Question #2 : Derivatives

The position of an object is described by the following equation:

Find the acceleration of the object at  second.

Explanation:

Acceleration is the second derivative of position, so we must first find the second derivative of the equation for position:

Now we can plug in t=1 to find the acceleration of the object after 1 second:

### Example Question #15 : How To Find Acceleration

The velocity of an object is given by the following equation:

Find the acceleration of the object at  seconds.

Explanation:

Acceleration is the derivative of velocity, so we must take the derivative of the given equation to find an equation for acceleration:

Now we can plug in t=2 to find the acceleration of the object at 2 seconds:

### Example Question #16 : How To Find Acceleration

The position vector of an object moving in a plane is given by . Find the object's acceleration when .

Explanation:

To find the acceleration, we must differentiate the position vector twice. Differentiating the position vector once gives the velocity vector:

Differentiating the second time gives acceleration:

Using 3 as the value for  gives

### Example Question #17 : How To Find Acceleration

If the position of a particle is: , What is the acceleration at ?

Explanation:

To find the acceleration equation of the particle, we can differentiate the position equation twice.

Differentiating once gives the velocity equation:

Differentiating again give the acceleration equation:

Using 3 as the value for ,

### Example Question #18 : How To Find Acceleration

What is the acceleration at time  when the acceleration is given by  is time in seconds.

Explanation:

To find the acceleration at time 4 seconds, we simply subsitute in 4 seconds into the acceleration function.

### Example Question #19 : How To Find Acceleration

The velocity function is .

What is the acceleration function?