Longer to Rise or to Fall?
Answer: Longer to fall, for Newton models with air resistance.
Example 3, Edwards-Penney 2.3
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A bolt is shot straight upward (along the gravity vector) with initial
velocity v1(0)=49 m/s from a crossbow at ground level.
Assume air resistance proportial to the square of the velocity with
drag rho=0.0011, giving the equations
(12) (d/dt)v1(t) = - 9.8 - 0.0011 v1(t)^2 [v1=rise]
(15) (d/dt)v2(t) = - 9.8 + 0.0011 v2(t)^2 [v2=fall]
Compare the rise and fall times for this Newton model with the models
(1) (d/dt)v3(t) = -9.8
[v3=velocity, no air resistance]
(4) (d/dt)v4(t) = - 9.8 - 0.04 v4(t)
[V4=velocity, linear air resistance]
Results
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Model Max Flight Rise Fall Impact
used Height time time time velocity
===== ====== ====== ==== ==== ========
v3 122.5 10.00 5.00 5.00 49.00
v4 108.28 9.41 4.56 4.85 43.23
v1 & v2 108.47 9.41 4.61 4.80 43.49