Question

given s(t) = t² + 3t, where s(t) is in feet and t is in seconds, find each of the following. a) v(t) b) a(t) c) the velocity and acceleration when t = 2 sec a) v(t) = □

Explanation:

Step1: Recall velocity as derivative of position

Velocity \( v(t) \) is the derivative of the position function \( s(t) \). The power rule for differentiation is \( \frac{d}{dt}(t^n) = nt^{n - 1} \).
For \( s(t)=t^{2}+3t \), we differentiate term - by - term.
The derivative of \( t^{2} \) with respect to \( t \) is \( 2t^{2 - 1}=2t \) (using the power rule with \( n = 2 \)).
The derivative of \( 3t \) with respect to \( t \) is \( 3\times1\times t^{1 - 1}=3 \) (using the power rule with \( n = 1 \)).

Step2: Combine the derivatives

\( v(t)=\frac{d}{dt}(s(t))=\frac{d}{dt}(t^{2}+3t)=\frac{d}{dt}(t^{2})+\frac{d}{dt}(3t)=2t + 3 \)

Answer:

\( 2t + 3 \)