Under the influence of various forces, the altitude h(t) of a rocket from its mo

Question

Under the influence of various forces, the altitude h(t) of a rocket from its moment of launch until its fall to Earth is given by the formula h(t)=-t^4+8t^3-3t^2+10t+375, with h in feet above sea level and t in seconds.

(a) From what altitude is the rocket launched?
(b) When does the rocket fall into the ocean?
(c) What is the rocket's maximum altitude?
(d) What is the rocket's maximum upward velocity?
(e) What is the rocket's maximum downward velocity?
(f) What is the rocket's maximum upward acceleration?
(g) What is the rocket's maximum downward acceleration?

Explanation / Answer

The altitude of a rocket from its moment of launch, until its fall to Earth which given by -

h (t) = - t4 + 8 t3 - 3 t2 + 10 t + 375

where, h = measured in feet above sea level

t = measured in second

(a) The rocket launched at t = 0.

(b) When h = 0, the rocket fall into the ocean.

(c) The rocket's maximum altitude which will be given as :

h (t) = - t4 + 8 t3 - 3 t2 + 10 t + 375

h (0) = - (0)4 + 8 (0)3 - 3 (0)2 + 10 (0) + 375

h (0) = 375 m

(d) The rocket's maximum upward velocity will be given as :

taking a derivative of above eq.

v = dh/dt = - 4 t3 + 24 t2 - 6 t + 10

at t = 1 sec, we have

vmax = - 4 (1)3 + 24 (1)2 - 6 (1)

vmax = 14 m/s

(f) The rocket's maximum upward acceleration which will be given as :

again, derivative of par-d equation -

a = dv / dt = - 12 t2 + 48 t - 6

at t = 1 sec

amax = - 12 (1)2 + 48 (1) - 6

amax = 30 m/s2

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