Velocity of projectile with both quadratic and constant resistive forceSuppose 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?

Question

Velocity of projectile with both quadratic and constant resistive forceSuppose 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?

Gallichi5mtwt · Accepted Answer

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  xDefining 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  xSubsequently,      m          2      α        ln  ⁡  u  =  −  x  +  CWe 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&#039;ll get u(p). From that you get v(p).

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

Answered question

Answer & Explanation

New Questions in Electric current