# Calculus 1 : How to find acceleration

## Example Questions

### Example Question #121 : How To Find Acceleration

A snowball rolls into a valley. Its position is represented by , where  represents distance in terms of meters and  represents time in terms of seconds.

What is the acceleration of the snowball?

Explanation:

Take the second derivative, using the Power Rule () twice, of :

### Example Question #121 : How To Find Acceleration

A weight hanging from a spring is stretched down 3 units beyond its rest position and released at time  to bob up and down. Its position at any later time  is

What is the acceleration at time ?

Explanation:

### Example Question #123 : How To Find Acceleration

The position of a particle is given by the expression

.

At what time  is the acceleration of the particle ?

Explanation:

To find the acceleration of a particle given the equation for position x(t), utilize the relationship

Therefore, to find the acceleration, we take the second derivative of x(t):

Now to find where the acceleration is 30, we set 6t equal to 30 and solve for t:

Which gives us the correct answer.

### Example Question #124 : How To Find Acceleration

The velocity of a particle is given by the equation

.

To two decimal places, what is the acceleration of the particle at  ?

Explanation:

The acceleration of a particle is given by the first derivative of the velocity:

Here, the chain rule must be used to fine the derivative of the velocity:

Finally, evaluate the expression for acceleration at t=3:

### Example Question #125 : How To Find Acceleration

Find  at , where

.

Explanation:

In order to evaluate at , we need to find .

We found this by using the definition of derivatives for exponential functions and the product rule.

Remember that the definition of derivaties for exponential functions are

.

Remember that the product rule is as follows:

Now lets find .

Now lets evaluate .

### Example Question #126 : How To Find Acceleration

Find the acceleration of a particle at  with a position defined by .

Explanation:

Acceleration of a particle is given by the second derivative of the position function. The position function is

Taking the first derivative gives the velocity function:

Taking the second derivative gives the acceleration function:

Evaluating the function at  gives the acceleration at this time:

### Example Question #127 : How To Find Acceleration

A given ball has a position defined by the equation . What is its acceleration at time ?

Explanation:

Acceleration is defined as the second derivative of position, or .

According to the Power Rule, the first derivative

for all ;

therefore, the second derivative

Applying this to

, we get

and

.

Plugging in

### Example Question #128 : How To Find Acceleration

A given propeller plane has a position defined by the equation . What is its acceleration at time ?

Explanation:

Acceleration is defined as the second derivative of position, or .

According to the Power Rule, the first derivative

for all ;

therefore, the second derivative

Applying this to

, we get

and

.

Plugging in

### Example Question #129 : How To Find Acceleration

A given object has a position defined by the equation . What is its acceleration at time ?

Explanation:

Acceleration is defined as the second derivative of position, or .

According to the Power Rule, the first derivative

for all ;

therefore, the second derivative

Applying this to

, we get  and .

Plugging in

### Example Question #130 : How To Find Acceleration

If the position of particle is , then what is its acceleration function ?