I have a question over the calculation of air resistance. When you're calculating the air resistance of a bullet, it's velocity is decreasing over time. The air resistance is also dependent on the velocity and thus, the air resistance is decreasing as the velocity decreases. Overall, the velocity is dependent on the acceleration which is dependent on the velocity. Is there a way to overcome this without using multivariable calculus?

Question

I have a question over the calculation of air resistance. When you&#039;re calculating the air resistance of a bullet, it&#039;s velocity is decreasing over time. The air resistance is also dependent on the velocity and thus, the air resistance is decreasing as the velocity decreases. Overall, the velocity is dependent on the acceleration which is dependent on the velocity. Is there a way to overcome this without using multivariable calculus?

Alannah Yang · Accepted Answer

In a problem where the only force is air resistance (neglecting gravity), the (one-dimensional) equation of motion is  F  =  m            x      ¨        =  f  (  v  )  ,where the       |    f  (  v  )      |   is the magnitude of the velocity-dependent aerodynamic friction, and the sign of   f  (  v  ) is always opposite that of v, so that the friction works to decrease the speed.Rewriting   v  =            x      ˙      , the equation becomes  m            v      ˙        =  m            d      v              d      t        =  f  (  v  )  ,which is a first-order, separable, ordinary differential equation. The solution can be reduced to performing integrals and inverting the resulting functions. An analytic solution may be found if the nontrivial integral in  m  ∫            d      v              f      (      v      )        =  ∫  d  t  +  Ccan be done in closed form. This yields v as a function of t, and v(t) can be integrated one more time to give x(t).

Calculation of air resistance. Is there a way to overcome this without using multivariable calculus?

Answered question

Answer & Explanation

New Questions in Advanced Physics