Velocity of projectile with both quadratic and constant resistive force Suppose we have a projectil

Nubydayclellaumvcd 2022-05-14 Answered
Velocity of projectile with both quadratic and constant resistive force
Suppose we have a projectile of mass m that impacts a material with velocity v 0 . As it travels through the material it is subject to the resistive force F = α v 2 + β. How can I determine v ( t ), the velocity of the projectile at time t where t=0 is the impact time, and v ( p ), the velocity of the projectile when it reaches a penetration depth of p?
This question addresses v(t) for an exclusively quadratic resistive force ( β = 0), but is there a more general formula for when there is a constant component?
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Answers (1)

Gallichi5mtwt
Answered 2022-05-15 Author has 18 answers
Using v d v = a d x where a = F / m = α m v 2 β m we have
v d v = ( α m v 2 β m ) d x v α m v 2 + β m d v = d x
Defining the new variable u = α m v 2 + β m we have d u d v = 2 α m v. Thus, v d v = m 2 α d u and the integral reduces to
m 2 α 1 u d u = d x
Subsequently,
m 2 α ln u = x + C
We use the initial condition that for x=0, v = v 0 and u = u 0 = α m v 0 2 + β m . Hence C = m 2 α ln u 0 . With some straightforward algebra you should be able to get
u ( x ) = u 0 exp ( 2 α m x )
Let x=p and you'll get u(p). From that you get v(p).
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