Gary Novak

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Complete Math

Velocities

From Falling Object page.
Symbols are given on Falling Object page.

Formulas: v = gt

t =
 √ 2s/g
(from s = ½at²)

g = 9.81 m/s (rounded throughout)

Numbers for each object:
A - 4 kg dropped 1 m
B - 1 kg dropped 4 m
C - 4 kg dropped 1 sec
D - 1 kg dropped 4 sec

Time Dropping:
A - 0.45152364098573 sec
B - 0.90304728197146 sec
C - 1 sec
D - 4 sec

A: t = [2(1m)/9.81]½ = 0.45152364098573 sec
B: t = [2(4m)/9.81]½ = 0.90304728197146 sec

Velocity Dropping:
A - 4.42944691807002 m/sec
B - 8.85889383614004 m/sec
C - 9.8100000000000 m/sec
D - 39.2400000000000 m/sec

A: v = 9.81(0.45152364098573) = 4.42944691807002 m/s
B: v = 9.81(0.90304728197146) = 8.85889383614004 m/s
C: v = 9.81(1) = 9.8100000000000 m/sec
D: v = 9.81(4) = 39.2400000000000 m/sec

Constant C in burn time:
A - 7783.22401633603
B - 7649.69262371151
C - 7783.22401633603
D - 7649.69262371151

velocity of rocket =
-103ln(100mo - t) + c

-c = velocity at t=0

c = 103ln(100m0)

A: c = 103ln[100(24)] = 7783.22401633603
B: c = 103ln[100(21)] = 7649.69262371151
C: c = 103ln[100(24)] = 7783.22401633603
D: c = 103ln[100(21)] = 7649.69262371151

Total Time of Burn:
A - 10.6071633272070 sec
B - 18.5215158540212 sec
C - 23.4288933861318 sec
D - 80.8081749880098 sec

t = mo/0.01 - [(c-v)/103]ex

(ex is inverse natural log for preceding quantity.)

A: 24kg/0.01 - [(7783.22401633603-4.42944691807002)/103]ex = 10.6071633272070 sec

B: 21kg/0.01 - [(7649.69262371151-8.85889383614004)/103]ex = 18.5215158540212 sec

C: 24kg/0.01 - [(7783.22401633603-9.81)/103]ex = 23.4288933861318 sec

D: 21kg/0.01 - [(7649.69262371151-39.24)/103]ex = 80.8081749880098 sec

mp = mass of payload = 1 kg or 4 kg
tp = time for payload including mass ratio integrated with time

equals test mass divided by average total mass (loses 0.01kg/s).

The mass ratio integrated with time is this:

mp/(mo - 0.01t) dt =

-100mpln(100mo - t) + c

-c is quantity at t = 0

c = 100mpln(100m0)

Constant C in Mass Fraction:
A - 3113.28960653441
B - 764.969262371151
C - 3113.28960653441
D - 764.969262371151

A: c = 100(4kg)ln[100(24kg)] = 3113.28960653441
B: c = 100(1kg)ln[100(21kg)] = 764.969262371151
C: c = 100(4kg)ln[100(24kg)] = 3113.28960653441
D: c = 100(1kg)ln[100(21kg)] = 764.969262371151

Time of Burn for Payload Only:
A - 1.77177876722800 sec
B - 0.885889383614004 sec
C - 3.92400000000000 sec
D - 3.92400000000000 sec

A: t = -100(4kg)ln[100(24kg) - 10.6071633272070] + 3113.28960653441 = 1.77177876722800 sec

B: t = -100(1kg)ln[100(21kg) - 18.5215158540212] + 764.969262371151 = 0.885889383614004 sec

C: t = -100(4kg)ln[100(24kg - 23.4288933861318] + 3113.28960653441 = 3.92400000000000 sec

D: t = -100(1kg)ln[100(21kg - 80.8081749880098] + 764.969262371151 = 3.92400000000000 sec

Ratios:
A/B – 2.00000000000000
C/D – 1.00000000000000

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