Answer:
Explanation:
check the pictures attached to further understand and i hope it works
Answer:
[tex]v = \frac{-384}{5} (1 - \frac{12}{5} e^{ -5t/12})[/tex]
Explanation:
Weight of the firefighter, W = 192 lb
W = mg
g = 32 ft/s²
Mass of the firefighter, m = W/g
m = 192/32
m = 6 slugs
k = 2.5 lb-s/ft
The force, F = ma = kv
2.5v = 6a
a = 2.5v/6
a = 5v/12
The fundamental dynamic equation;
dv/dt + drag + gravity = 0
dv/dt = -g-a
dv/dt = -32-5v/12..............(a)
The motion will attain terminal velocity when dv/dt = 0
-32 - 5v/12 = 0
-32 = 5v/12
-384 = 5v
v₀ = -384/5
v₀ = -384/5
dv/dt = -32 - 5v/12
[tex]\frac{dv}{-32 -5v/12} = dt[/tex]
[tex]-12/5 ln(32 + 5v/12) = t + c\\ln(32 + 5v/12) = -5t/12 + lnc\\ln(32 + 5v/12) - ln c = -5t/12\\ln\frac{32 + 5v/12}{c} = -5t/12[/tex]
Take exponential of both sides
[tex]\frac{32 + 5v/12}{c} =e^{ -5t/12}\\32 + 5v/12 = ce^{ -5t/12}\\5v/12 = -32 + ce^{ -5t/12}\\v = 12/5 (-32 + ce^{ -5t/12})\\[/tex]
c = v₀ = -384/5
[tex]v = 12/5 (-32 - \frac{384}{5} e^{ -5t/12})\\v = \frac{-384}{5} (1 - \frac{12}{5} e^{ -5t/12})[/tex]
