CPM Homework Help : CALC Problem 12-83

Multiple Choice: A particle moves in the plane according to the equations $x(t) =\operatorname{cos} πt$ and $y ( t ) = \frac { t ^ { 2 } } { t + 1 }$ . What is the magnitude of the velocity vector at $t = \frac { 1 } { 2 }$ ?

$\frac { 1 } { 6 }$

$\pi + \frac { 5 } { 9 }$

$\frac { \sqrt { 106 } } { 9 }$

$\frac { \sqrt { 81 \pi ^ { 2 } + 25 } } { 9 }$

Step 1:

$\vec{v}(0.5)=\langle -\pi\sin(\pi/2),\frac{0.5^2+1}{(1.5)^2}\rangle=\langle -\pi,\frac{5}{9}\rangle$

Step 2:

Sketch this vector, then use the Pythagorean Theorem to determine its magnitude.