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September 30, 2014

September 30, 2014

Posted by **Michael** on Friday, February 22, 2008 at 4:06pm.

--------( 20e^(-t/2), for 0 <= t <= 10 seconds)

a(t) = (

--------( -16, for t > 10 seconds)

a) At what time does the rocket begin to descend?

b) How high does the rocket reach?

c) What is the velocity when the rocket impacts the earth?

d) Write a formula for the position of the rocket with respect to time for t > 10 seconds.

Show all work. Thanks a lot!

- Calc -
**drwls**, Friday, February 22, 2008 at 4:24pmNo, YOU show your work.

a) It begins to descend when V=0. Integrate the acceleration until time t when the integral is zero

b) Integrate the V(t) equation from (a) to get altitude Y vs t. Plug in the t you got in part (a)to get maximum height.

(c) Knowing the maximum height H, you can get the velocity after falling to earth from the equation

V = sqrt (2 a H)

(d) This should have been done while doing part (b)

This problem seems to be using an incorrect value for the acceleration "a" after 10 s. It should be 32 ft/s^2. Use the value they gave you, anyway. Also, for the first 10 seconds, the acceleration should be increasing, not decreasing, because the rocket's mass decreases with time during the time the engine is on. Whoever assigned the problem could use a refresher course in rockets

- Calc -
**Damon**, Friday, February 22, 2008 at 4:41pmHow high and how fast at 10 seconds first:

v = integral a dt + constant c

v(t) = 0 + 20 int dt e^-(t/2)

= 20 (-2) e^-(t/2) + c

= -40 e^-(t/2) + c

when t = 0, v = 0 so

0 = -40 + c

c = 40

so

v = 40(1-e^(-t/2) )for the first ten seconds

at ten seconds

v(10) = 40(1-.00674) = 39.7 m/s

h = int v dt + c2

h = 40 int (1 dt) -10 int (e^-(t/2))dt + c2

h = 40 t +80 e^(-t/2) + c2

when t = 0, h = 0

so c2 = -80

so

h = 40 t + 80 e^-(t/2) -80

at t = 10 seconds

h = 400 + .539 -80 = 321 m

SO

at ten seconds

Vo = 39.7 up

and

h = 321 meters up

Those are your initial conditions for the constant acceleration down phase. I think you can take it from there.

CHECK MY ARITHMETIC!!!

- Calc -
**Damon**, Friday, February 22, 2008 at 4:44pmI mean feet, not meters.

Yes, acceleration is crazy. It should increase during the propulsion phase and it should be g, 32 ft/s^2 down, after the engine cuts out. I assume this is math class and certainly not a physics class :)

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