Suppose that a model rocket blasts off, straight up, at time t = 0, with a vertical displacement given by a function f (t) for t ≥0. During the first two seconds of the launch (0 ≤t ≤2), the engine is burning, and the rocket is accelerating upward with a constant second derivative f ′′(t) = 40 (metres/sec2). At t = 2, the engine stops burning, and gravity takes over, such that the rocket is then accelerating downward with a constant second derivative of f ′′(t) = −10 (metres/sec2) for all t ≥2. (i) Assume that •The rocket begins (at time t = 0) with a vertical displacement of zero (f (0) = 0). •The rocket begins (at time t = 0) with a vertical velocity of zero (f ′(0) = 0). •The vertical displacement f (t) is continuous for all t ≥0. •The vertical displacement f (t) is differentiable for all t ≥0. Solve for the vertical displacement f (t) of the rocket for all t ≥0. (NOTE: Your solution should be a piecewise function of t, and you do not need to worry about what happens to the rocket after it hits the ground). (ii) What is the maximum height achieved by the rocket (in metres)? Justify your answer. (iii) At what time (in seconds) does the rocket hit the ground? Approximate your answer to two decimal places.
Suppose that a model rocket blasts off, straight up, at time t = 0, with a vertical displacement given by a
function f (t) for t ≥0. During the first two seconds of the launch (0 ≤t ≤2), the engine is burning, and
the rocket is accelerating upward with a constant second derivative f ′′(t) = 40 (metres/sec2). At t = 2,
the engine stops burning, and gravity takes over, such that the rocket is then accelerating downward
with a constant second derivative of f ′′(t) = −10 (metres/sec2) for all t ≥2.
(i) Assume that
•The rocket begins (at time t = 0) with a vertical displacement of zero (f (0) = 0).
•The rocket begins (at time t = 0) with a vertical velocity of zero (f ′(0) = 0).
•The vertical displacement f (t) is continuous for all t ≥0.
•The vertical displacement f (t) is differentiable for all t ≥0.
Solve for the vertical displacement f (t) of the rocket for all t ≥0.
(NOTE: Your solution should be a
what happens to the rocket after it hits the ground).
(ii) What is the maximum height achieved by the rocket (in metres)? Justify your answer.
(iii) At what time (in seconds) does the rocket hit the ground? Approximate your answer to two decimal
places.
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