Parametric Equations Question 3

calculator allowed For time $t\gt 0$ , the position of an object moving in the $xy$ -plane is given by the parametric equations $x(t)=t\cos{\left(\dfrac{t}{2}\right)}$ and $y(t)=\sqrt{t^2+2t}$ . What is the speed of the object at time $t=1$ ?

1.155

1.319

1.339

1.810

Approach

The magnitude or speed is given by:

|v(t)|=\sqrt{v_x^2+v_y^2}

x(t)=t\cos{\left(\dfrac{t}{2}\right)}

v_x=x'(t)=-\frac{t}{2}\sin{\frac{t}{2}}+\cos{\frac{t}{2}}

x'(1)\approx 0.638

y(t)=\sqrt{t^2+2t}

v_y=y'(t)=\frac{2t+2}{2\sqrt{t^2+2t}}

y'(1)\approx 1.155

|v(t)|=\sqrt{v_x^2+v_y^2}

|v(1)|=\sqrt{(x'(1))^2+(y'(1))^2}

\approx \boxed{1.319}