A spacecraft’s trajectory in deep space is described analytically by the position…

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

A spacecraft’s trajectory in deep space is described analytically by the position vector

\mathbf r(u)=\langle e^{2u},\,u^{3},\,\sqrt{1+u}\rangle\text{ km},

where the scalar parameter

u

depends on mission time

t

(in hours) according to

u(t)=t^{2}+1

. Assume all functions are differentiable for

t>0

. Throughout this problem, express all vector answers in component form and include appropriate units. (a) Derive an explicit formula for the spacecraft’s velocity vector

\mathbf v(t)=\dfrac{d\mathbf r}{dt}

. Simplify each component as much as possible. (b) Obtain an explicit expression for the acceleration vector

\mathbf a(t)=\dfrac{d^2\mathbf r}{dt^2}

. Clearly indicate the differentiation steps leading to each component. (c) Evaluate

\mathbf v(2)

and

\mathbf a(2)

. State numerical values to three decimal places. (d) Compute the magnitude of the velocity at

t=2

and interpret the result in the context of the spacecraft’s motion.

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