A television camera is located on level ground from the base of a vertical launch tower

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

Problem

A television camera is located on level ground

4000\,\text{ft}

from the base of a vertical launch tower. The camera is pointed at a rocket that rises vertically along the tower and then continues upward on the same vertical line. At time

t

seconds after lift-off, the rocket’s height above ground level is given by

h(t)

ft and the angle

\theta(t)

radians is measured between the ground and the camera’s line of sight to the rocket. When

h = 3000\,\text{ft}

, the rocket’s upward velocity is

600\,\text{ft/s}

and its upward acceleration is

50\,\text{ft/s}^2

. Assume that

h(t)

is twice differentiable for the time interval considered. (a) Derive an expression for the instantaneous rate of change

\frac{d\theta}{dt}

in terms of

h

and

\frac{dh}{dt}

. (b) Evaluate

\frac{d\theta}{dt}

when

h = 3000\,\text{ft}

and

\frac{dh}{dt} = 600\,\text{ft/s}

. Include appropriate units. (c) Interpret the sign and magnitude of the value found in part (b) within the physical context of the problem. (d) Using your result from part (a), obtain an expression for the angular acceleration

\frac{d^2\theta}{dt^2}

in terms of

h

\frac{dh}{dt}

, and

\frac{d^2h}{dt^2}

. Then determine

\frac{d^2\theta}{dt^2}

at the instant described above. State units and indicate whether the camera’s rotation is speeding up or slowing down.

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