Homework Help: Calc

Posted by Michael on Sunday, February 24, 2008 at 4:55pm.

A rocket is launched with an initial velocity of zero, and with acceleration in feet per second per second defined by:

--------( 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.

Thanks!

Calc - Damon, Sunday, February 24, 2008 at 6:21pm
I believe both drwls and I did this problem not long ago.
It is kind of wacky.
The acceleration should increase, not decrease, while the engine is on for the first ten seconds because the mass decreases as fuel is used up.
The acceleration down during the ballistic phase with the engine off should be -32 ft/s^2, gravity, not -16.
Never the less I will get you started.
a = d^2h/dt^2 = 20 e^-t/2
v = dh/dt = c - 40 e^-(t/2)
since v = 0 at t = 0, c = 40 and
v = 40 (1-e^-(t/2))
Now we can get v at t = 10
v = 39.7 ft/s at t = 10
how high is it at t = 10?
h = c2 + 40 t + 80 e^(-t/2)
h = 0 when t = 0 so c2 = -80
h = 40 t +80(e^-(t/2) -1)
now when t = 10
h = 400 + 80 (.006-1) = 320
so NOW we have a brand new problem at t = 10 seconds
a = -16
Vo = 39.7 ft/s
Ho = 320 ft
Now
v = vo - at
at the top, v = 0
0 = 39.7 - 16 t
t = 2.48 s

SO part a)
t+the original ten seconds = 12.48 s

h = Ho + Vo t + (1/2) a t^2
h = 320 + 39.7 * 2.48 -8 *2,48^2
h = 320 + 98.5 - 49.2

So part b)
h = 369 feet at the top.

c it falls from 369 feet
v = 0 - 32 t

-369 = (1/2)(-32) t^2
-369 = -16 t^2
t = 4.80 seconds falling
SO
Part c
v at ground = -32(4.8) = - 154 ft/s

I can not do part d because they had g wrong, 16 instead of 32 during the ballistic phase.
If it were -32, as it should be, then
h = Ho + Vo (t-10) + (1/2)(-32) (t-10)^2
Ho is 320
Vo is 39.7 If I did the first ten seconds right
I used t-10 because I assume they mean total t since liftoff, not time after ten seconds.

Answer this Question

Related Questions

Calc - A rocket is launched with an initial velocity of zero, and with ...
algebra - If a rocket is launched with an initial velocity of 1600 feet per ...
math - a launched rocket has an altitude, in meters, given by the polynimial h+...
Algbrah - A launched rocket has an altitude in meters, given by the polynomial h...
math 117 - A launched rocket has an altitude in meters, given by the polynomial ...
math 117 - A launched rocket has an altitude in meters, given by the polynomial ...
math 116 - A launched rocket has an altitude in meters, given by the polynomial ...
math 117 - A launched rocket has an altitude in meters, given by the polynomial ...
math-algebra - A launched rocket has an altitude, in meters, given by the ...
Algebra~Scary problem! - A launched rocket has an altitude, in meters, given by ...

For Further Reading

First Name:
School Subject:
Answer: