Question

How can I use derivatives to find acceleration, given a position-time function?

Answer 1

Hint: We know that we use the differentiation to find the rate of change of a quantity with time. We know that the derivative of the displacement is called the velocity. When we differentiate velocity, we will get acceleration.

Complete step by step solution:
We are asked to find the acceleration when the position-time function is given.
We know that the acceleration is the rate of change of velocity with respect to time. And the velocity is the rate of change of displacement with respect to time.
Suppose that the position-time function or displacement is given by $x\left(t\right).$
We can find the velocity of the function by finding the first derivative of the displacement with respect to time.
So, the velocity is $v\left(t\right)={\displaystyle \frac{dx\left(t\right)}{dt}}.$
That is, the rate of change of the position-time function is the velocity.
Now we need to find the acceleration. We can find the acceleration by differentiating the velocity $v\left(t\right)$ with respect to time.
So, we will get the acceleration as $a\left(t\right)={\displaystyle \frac{dv\left(t\right)}{dt}}.$
Therefore, the acceleration can be found by differentiating the velocity with respect to time while the velocity can be found by differentiating the given position-time function with respect to time.

Hence the acceleration is $a\left(t\right)={\displaystyle \frac{dv\left(t\right)}{dt}}.$

Note: We can learn that the acceleration can be found by differentiating the given position-time function twice. That is, the acceleration is the second derivative of the position-time function.