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

Nubydayclellaumvcd

Nubydayclellaumvcd

Answered question

2022-05-14

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?

Answer & Explanation

Gallichi5mtwt

Gallichi5mtwt

Beginner2022-05-15Added 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).

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?