How Do You Calculate Terminal Velocity Involving Drag?

In summary, the conversation discusses the process of deriving an equation for the velocity of a falling body with respect to terminal velocity. The equation takes into account the influence of drag on the speed. The individual asking the question shares the steps they took to derive the equation and asks for help regarding a mistake in the integration part. After receiving assistance, they realize their mistake and express gratitude.
  • #1
spacetimedude
88
1
Hello PF,
I have once simple (well, not so simple for me) question.

I'm trying to derive an equation for the velocity of a falling body with accordance to terminal velocity.

The equation incorporates drag proportional to the speed.

m*dv/dt=mg-bv

and

mg/b=terminal velocity vt

So the steps I took were:

m*dv/dt+bv=mg

(m/b)*(dv/dt)+v=vt

dv/dt=(b/m)(vt-v)

dv/(vt-v)=(b/m)dt

Integrating both sides would give
ln[(vt-v)/(vt)]=(b/m)t

But the textbook says that I'm supposed to get negative (b/m)t on the left side.

Have I made a mistake on the integration part?

Any help will be deeply appreciated.
 
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  • #2
hmm. I seem to have gotten the answer if I just divided the entire equation by -b in the beginning without bringing the bv to the left side. Have I made a mistake on the integration part?
 
  • #3
You just missed the minus sign while integrating (chain rule)
 
  • #4
pshh. I can't believe I missed that. Thanks so much king vitamin!
 
  • #5


Hello,

Thank you for your question. It appears that you are on the right track with your derivation. The equation you have derived is known as the differential equation for terminal velocity, and the solution you have found is correct. However, the textbook may be referring to the final form of the equation, which involves taking the natural logarithm of both sides and then rearranging the terms to isolate the velocity. This would result in a negative term for the coefficient of time, as you mentioned.

To clarify, here is the full derivation:

m*dv/dt + bv = mg

Dividing both sides by m:

dv/dt + (b/m)*v = g

This is a first-order linear differential equation, which can be solved using the integrating factor method. The integrating factor is e^(b/m*t), so multiplying both sides by this factor:

e^(b/m*t)*(dv/dt) + (b/m)*e^(b/m*t)*v = e^(b/m*t)*g

Applying the product rule on the left side, we get:

d/dt(e^(b/m*t)*v) = e^(b/m*t)*g

Integrating both sides with respect to time:

∫ d/dt(e^(b/m*t)*v) dt = ∫ e^(b/m*t)*g dt

Using the fundamental theorem of calculus, the left side becomes e^(b/m*t)*v, and the right side becomes e^(b/m*t)*(g*t + C). So we have:

e^(b/m*t)*v = e^(b/m*t)*(g*t + C)

Dividing both sides by e^(b/m*t), we get:

v = g*t + C

To find the value of C, we can use the initial condition that at t=0, v=0. So plugging in these values, we get C=0. Therefore, the final equation for velocity as a function of time is:

v = g*t

To find the terminal velocity, we set the drag force equal to the weight of the object, as you did in your derivation:

mg = bv

Solving for v, we get:

v = mg/b = vt

Substituting this back into our original equation:

dv/dt + (b/m)*vt = g

Integrating both sides with respect to time:

∫ dv/dt dt + (b/m)*vt^2 =
 

FAQ: How Do You Calculate Terminal Velocity Involving Drag?

1. What is terminal velocity?

Terminal velocity is the maximum velocity that an object can reach when falling through a fluid, such as air or water. At this velocity, the object is no longer accelerating and experiences a constant velocity.

2. How is terminal velocity calculated?

Terminal velocity can be calculated using the equation v = √(2mg/ρAC), where v is the terminal velocity, m is the mass of the object, g is the acceleration due to gravity, ρ is the density of the fluid, A is the cross-sectional area of the object, and C is the drag coefficient.

3. What factors affect terminal velocity?

The factors that affect terminal velocity include the mass and shape of the object, the density and viscosity of the fluid, and the presence of external forces such as wind or air resistance. Additionally, the altitude and temperature of the surrounding environment can also impact terminal velocity.

4. How does air resistance affect terminal velocity?

Air resistance, also known as drag force, increases as the speed of an object increases. At low speeds, the drag force is small and the object accelerates. However, as the object approaches terminal velocity, the drag force becomes equal to the force of gravity, resulting in a constant velocity.

5. Can terminal velocity be reached in a vacuum?

No, terminal velocity can only be reached when there is a fluid present to create a drag force. In a vacuum, there is no air or other fluid to create resistance, so an object will continue to accelerate until it reaches its maximum potential velocity.

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