A research rocket is launched vertically from a pad that is 2 km from a tracking station

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

A research rocket is launched vertically from a pad that is 2 km from a tracking station. The station’s sensor measures

\theta(t)

, the angle of elevation (in radians) to the rocket, and the sensor’s electronics can determine

\dfrac{d\theta}{dt}

directly. At a certain instant when the angle of elevation is

\dfrac{\pi}{4}

, the sensor reports that

\dfrac{d\theta}{dt}=0.015\text{ rad/s}

. (a) Without doing any calculations, describe qualitatively how

\theta

and

\dfrac{d\theta}{dt}

are expected to change as the rocket ascends. (b) Derive an expression for the rocket’s vertical velocity

v=\dfrac{dh}{dt}

in terms of

\theta

and

\dfrac{d\theta}{dt}

, where

h

is the rocket’s height in kilometers. (c) Use your formula from part (b) to determine the rocket’s velocity (in m/s) at the given instant. (d) Interpret the meaning of the sign of

\dfrac{d\theta}{dt}

and of

v

in this context.

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