# Calculus 1 : How to find acceleration

## Example Questions

### Example Question #51 : How To Find Acceleration

The velocity of an object is given by the equation . What is the acceleration of the object at time ?

None of these.

Explanation:

The acceleration of the object can be found by differentiating the velocity equation of the object. To do this we can use the product rule where if

.

We must also use the power rule where if

.

Rewriting the velocity equation to , we can apply these rules to the velocity equation along with the knowledge that the derivative of  is  giving us,

.

To find the acceleration at  we now substitute the value of  for  in the acceleration equation.

### Example Question #52 : How To Find Acceleration

The velocity of an object is given by the equation . What is the initial acceleration of the object?

None of these.

None of these.

Explanation:

The initial acceleration of the object can be found by differentiating the object's velocity equation  and solving for .

To differentiate the velocity equation, we can use the quotient rule where if

.

Additionally, to find the derivative of the equation we must use the power rule where if

.

Applying these rules with the knowledge that the derivative of  is , we can find the acceleration equation.

The acceleration of the object is the value of the acceleration equation when .

Therefore the initial acceleration of the object is undefined.

### Example Question #53 : How To Find Acceleration

The jerk of an object is . What is the acceleration equation of the object if the acceleration is  at time ?

Explanation:

The acceleration equation can be found by integrating the jerk equation .

The integral of  is .

Therefore the acceleration equation is,

.

We can solve for the value of  by using the acceleration given to us at .

Therefore, .

The acceleration equation is then,

.

### Example Question #54 : How To Find Acceleration

The position of an object is given by the equation . What is the initial acceleration of the object?

Explanation:

The acceleration of the object can be found by differentiating the position equation  twice.

To differentiate this equation we can use the chain rule where if

.

Therefore the first derivative of the position equation, the velocity equation, is

.

Differentiating the equation a second time yields the acceleration equation.

The initial acceleration of the object is the acceleration of the object at time .

### Example Question #55 : How To Find Acceleration

The velocity equation of an object is . What is the acceleration equation of the object?

Explanation:

The acceleration equation of the object is the derivative of the velocity equation. To differentiate the velocity equation we can use the power rule where

.

Rewriting the velocity we obtain

.

Applying the power rule to the velocity equation gives us

.

Therefore, the acceleration equation is

.

### Example Question #56 : How To Find Acceleration

The position of an object is given by the equation . What is the equation of acceleration of this object?

Explanation:

The equation of acceleration can be found by differentiating the position equation twice. To differentiate the position equation we can use the power rule and the chain rule where if

and where if

Applying these two rule to the position equation, we can find the velocity equation

To find the acceleration equation we must differentiate the velocity equation. To do this we must use the product rule for the third term in the velocity equation. The product rule is where if

Using this rule we can differentiate the velocity equation

### Example Question #57 : How To Find Acceleration

The position of a boat is defined by the equation . What is the boat's acceleration at

Explanation:

The acceleration  is defined as the second derivative of its position , or .

Use the power rule to differentiate the position function twice.

Since  and .

Plugging in

### Example Question #58 : How To Find Acceleration

The position of a car is defined by the equation . What is the car's acceleration at ?

Explanation:

The acceleration  is defined as the second derivative of its position , or .

Use the power rule to differentiate.

Since  and .

Swapping in .

### Example Question #59 : How To Find Acceleration

The position of a particle is defined by the equation . What is the particle's acceleration at ?

Explanation:

The acceleration  is defined as the second derivative of its position , or .

Use the power rule to differentiate.

Since  and .

Swapping in .

### Example Question #60 : How To Find Acceleration

The position of an object is given by the equation . What is the acceleration of the object when ?