Chapter 4 - Integration - 4.7 Rectilinear Motion Revisited Using Integration - Exercises Set 4.7 - Page 330: 17

\begin{aligned} \int v(t) d t=-\frac{2}{\pi} \cos \frac{\pi t}{2}+C=&\text { (a) } s(t)\\ &s(0)=-\frac{2}{\pi} \cos 0+C=-\frac{2}{\pi}+C \Rightarrow \frac{2}{\pi}=C\\ &s(1)=-\frac{2}{\pi} \cos \frac{\pi}{2}+\frac{2}{\pi}=\frac{2}{\pi}\\ &v(1)=\sin \frac{\pi}{2}=1\\ &\text { Speed }=|v(1)|=1\\ &a(t)=v^{\prime}(t)=\frac{\pi}{2} \cos \frac{\pi t}{2}\\ 0=\frac{\pi}{2} \cos \frac{\pi}{2}=&a(1) \end{aligned} \begin{aligned} &\text { (b) } v(t)=\int a(t) d t=-\frac{3 t^{2}}{2}+C\\ &v(0)=0+C=0 \Rightarrow 0=C\\ &s(t)=\int v(t) d t=-\frac{t^{3}}{2}+C_{2}\\ &s(0)=0+C_{2}=1 \Rightarrow C_{2}=1\\ &s(1)=1-\frac{1}{2}=\frac{1}{2}\\ -\frac{3}{2}=&v(1)\\ &\text { Speed }=|v(1)|=\frac{3}{2}\\ -3=&a(1)\\ \end{aligned}

\begin{aligned} \int v(t) d t=-\frac{2}{\pi} \cos \frac{\pi t}{2}+C=&\text { (a) } s(t)\\ &s(0)=-\frac{2}{\pi} \cos 0+C=-\frac{2}{\pi}+C \Rightarrow \frac{2}{\pi}=C\\ &s(1)=-\frac{2}{\pi} \cos \frac{\pi}{2}+\frac{2}{\pi}=\frac{2}{\pi}\\ &v(1)=\sin \frac{\pi}{2}=1\\ &\text { Speed }=|v(1)|=1\\ &a(t)=v^{\prime}(t)=\frac{\pi}{2} \cos \frac{\pi t}{2}\\ 0=\frac{\pi}{2} \cos \frac{\pi}{2}=&a(1) \end{aligned} \begin{aligned} &\text { (b) } v(t)=\int a(t) d t=-\frac{3 t^{2}}{2}+C\\ &v(0)=0+C=0 \Rightarrow 0=C\\ &s(t)=\int v(t) d t=-\frac{t^{3}}{2}+C_{2}\\ &s(0)=0+C_{2}=1 \Rightarrow C_{2}=1\\ &s(1)=1-\frac{1}{2}=\frac{1}{2}\\ -\frac{3}{2}=&v(1)\\ &\text { Speed }=|v(1)|=\frac{3}{2}\\ -3=&a(1)\\ \end{aligned}