Physics problem; Find expression for velocity with varying acceleration

Hi. It's been 25 years since my last calculus class and I'm beyond rusty. I hope somebody her might be able to lend me a hand.

I'm working on a machine that discharges spherical solids horizontally in still air at a known speed. I'd like to study the sphere's speed over set time and/or distance intervals. The sphere will be slowed by aerodynamic drag. The standard drag formula D=.5 p (V^2) Cd A [where D is drag force, p is air density, V is velocity, Cd is coefficient of drag, and A is cross sectional area of sphere] can be used to determine the force that decelerates the sphere. As the force tending to decelerate the sphere is a function of velocity, it constantly decays after the sphere is discharged from the machine.

So I'm looking for an expression for V in terms of distance and an expression for V in terms of time.

I hope I've explained what I'm trying to do well enough. Let me know if I can clear up anything. Thanks.

Dave

Re: Physics problem; Find expression for velocity with varying acceleration

Quote:

Originally Posted by

**Wayintense** Hi. It's been 25 years since my last calculus class and I'm beyond rusty. I hope somebody her might be able to lend me a hand.

I'm working on a machine that discharges spherical solids horizontally in still air at a known speed. I'd like to study the sphere's speed over set time and/or distance intervals. The sphere will be slowed by aerodynamic drag. The standard drag formula D=.5 p (V^2) Cd A [where D is drag force, p is air density, V is velocity, Cd is coefficient of drag, and A is cross sectional area of sphere] can be used to determine the force that decelerates the sphere. As the force tending to decelerate the sphere is a function of velocity, it constantly decays after the sphere is discharged from the machine.

So I'm looking for an expression for V in terms of distance and an expression for V in terms of time.

I hope I've explained what I'm trying to do well enough. Let me know if I can clear up anything. Thanks.

Dave

(Borrowed liberally from Fowler's "Analytical Mechanics", 4 ed.)

This is going to be a bit strange if you haven't seen it before. I'll give you some guidance through the solution and have you fill in the steps as you go.

Initially, t = 0, v(0) = const. (That's the best we can do for an initial speed without more information.)

I am going to neglect the weight of the object (so no mg in the problem) and I'm going to assume that the force on the sphere is going to be strictly in the horizontal direction as I am also going to assume that the drag force is horizontal. You can work without these assumptions, but the problem gets really nasty really quickly.

So. We know that

$\displaystyle F(v) = m \frac{dv}{dt}$

$\displaystyle m \frac{dv}{dt} = - \frac{1}{2} \rho C_d A v^2$

This is a bit bulky so I'm going to define

$\displaystyle \alpha = \frac{\rho}{2}C_d A$

and our equation becomes

$\displaystyle m \frac{dv}{dt} = - \alpha v^2$

This is a separable differential equation. I'll leave the details here to you.

So doing the integration and a little cleaning up we have

$\displaystyle v = \frac{v_0}{\frac{\alpha v_0}{m} \cdot t + 1}$

(Recall the definition of $\displaystyle \alpha$ before you quote your answer.)

For the last part let us make a new definition:

$\displaystyle v = \frac{v_0}{kt + 1}$

Where

$\displaystyle k = \frac{\alpha v_0}{m}$

Then the rest falls into place:

x = dv/dt, integrate, solve this for t, then plug this into the equation above for t and there you have it.

Let's see how far you can go with this. Come back if you need more.

-Dan

Re: Physics problem; Find expression for velocity with varying acceleration

Hi Dan. Wow! Great help; thanks!

I'm stuck at the diferential equation, though. A friend of mine came up with a different expresion for v. I've been playing around with Wolfram Alpha and I can't replicate his expression or yours. If you wouldn't mind walking me through some of the intermediate dif eq seperation and integration steps, I'd greatly appreciate it. Thanks!

Dave

Re: Physics problem; Find expression for velocity with varying acceleration

Hi Dan.

Got it! Found my mistake.

Thnak you!

Dave