According to the question,
(a) Equation of the motion before parachute open,
ma=mg−0.76v
Equation of motion after parachute open,
ma′=mg−12v′
Now, For motion before parachute open,
mdtdv=mg−0.76v
∫(mg−0.76v)mdv=∫dt
Integrating both sides,
0.76−mln∣mg−0.76∣=t+C
applying condition that v(0)=0
C=0.76−mln∣mg∣
equation of the motion is given by,
0.76−mln∣mg−0.76∣=t+0.76−mln∣mg∣
0.76−mln∣mgmg−0.76∣=t
or v=0.76mg(1−e−m0.76t)
so velocity when parachute open,
putting values, mg = 174*32, m = 174 lb,
v=0.76174∗32(1−e−1740.76∗10)=306.4 ft/s
(b) for distance calculation,
dtdx=0.76mg(1−e−m0.76t)
∫0xdx=∫0100.76mg(1−e−m0.76t)dt
on integrating, we get,
x=0.76mg(t+0.76me−m0.76t)∣010
putting values, we get
x=0.76174∗32(10+0.76174e−1740.76∗10)=1556.8ft
(c) for critical value,
dtdv=0
so we obtain,
mg=0.76v
⟹v=0.76174∗32=7326.3 ft/s
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