Mathematics

Problem

A small research rocket is launched vertically upward from rest. The acceleration profile (positive upward) is described verbally below. • From

t = 0\,\text{s}

t = 4\,\text{s}

, the upward acceleration increases linearly from

0

+12\,\text{m/s}^2

. • From

t = 4\,\text{s}

t = 8\,\text{s}

, the acceleration decreases linearly to

-4\,\text{m/s}^2

(the engines throttle down past zero to provide a small downward thrust). • From

t = 8\,\text{s}

t = 12\,\text{s}

, the acceleration remains constant at

-4\,\text{m/s}^2

. • At

t = 12\,\text{s}

the engines shut off completely, after which the only acceleration is gravitational:

-9.8\,\text{m/s}^2

(assumed constant). Ignore air resistance throughout. (a) Determine the rocket’s velocity at

t = 4\,\text{s}

and at

t = 8\,\text{s}

. (b) Identify the rocket’s maximum upward velocity attained during the first

12\,\text{s}

of flight and state the time at which it occurs. (c) Calculate the rocket’s altitude (vertical displacement relative to the launch point) at

t = 12\,\text{s}

. (d) After engine cutoff, determine (i) the additional time required for the rocket to reach its highest point and (ii) that maximum altitude above the launch point.

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