Chapter 3.4, Problem 41E

To determine

To identify:The statement is true or false rate of change of the function.

Answer to Problem 41E

The statement saying acceleration is second derivative of position function is TRUE.

Explanation of Solution

Given information:

Need to verify if acceleration of a particle is the second derivative of the position function or not.

Lets say position function be represented by $s\left(t\right)$ .

As velocity of the particle is rate of change of position with respect to time $t$ , velocity function will be first derivative of position function defined as follows

  $v\left(t\right)=\frac{d}{dt}s\left(t\right)$   ......(1)

As acceleration of the particle is rate of change of velocity with respect to time $t$ , acceleration function will be first derivative of velocity function defined as follows

  $a\left(t\right)=\frac{d}{dt}v\left(t\right)$   ......(2)

Substitute value of $v\left(t\right)$ from (1) into (2).

  $a\left(t\right)=\frac{d}{dt}\left(\frac{d}{dt}s\left(t\right)\right)$

  $a\left(t\right)=\frac{d^2}{dt}\left(s\left(t\right)\right)$

Asacceleration that is $a\left(t\right)$ is second derivative of position function $s\left(t\right)$ , hence given statement is true.