Chapter 3.3, Problem 47E

To determine

To calculate: The value of $\frac{ds}{dt}$ and $\frac{d^{2}t}{d{t}^2}$ .

Answer to Problem 47E

The required values of $\frac{ds}{dt}$ and $\frac{d^{2}s}{d{t}^2}$ are $9.8t$ and 9.8 respectively.

Explanation of Solution

Given information:

The equation of: $s=4.9t^2$

Formula used:

Power rule of differentiation:

  $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$

Calculation:

The given equation is $s=4.9t^2$ .

Differentiate the given equation using the power rule $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$ .

  $\begin{array}{c}\frac{ds}{dt}=\frac{d}{dt}\left(4.9t^2\right)\\ \frac{ds}{dt}=2\left(4.9t^{2-1}\right)\\ \frac{ds}{dt}=9.8t\end{array}$

Differentiate the above derivative.

  $\begin{array}{c}\frac{d}{dt}\left(\frac{ds}{dt}\right)=\frac{d}{dt}\left(9.8t\right)\\ =9.8\frac{d}{dt}\left(t\right)\\ =9.8\left(1\right)\\ =9.8\end{array}$

Hence, the required values of $\frac{ds}{dt}$ and $\frac{d^{2}s}{d{t}^2}$ are $9.8t$ and 9.8 respectively.