How Can You Solve a Differential Equation Involving Exponential Drag Force?

[haruspex]

So, using the post equation in 19, I got 2, but using the 25post ,I get 1.
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I have no idea how you keep getting $e^{-c v_0} = e^{-c v_t}$. Where does that come from?
What I wrote in post #21 was, suppose we allow a constant of integration, k:
$e^{-c v_0} - e^{-c v_t}=\frac {-b c t} m+k$
If we evaluate that at t=0 we get:
$e^{-c v_0} - e^{-c v_0}=0+k$
Look carefully at the left hand side. Everything cancels there, leaving zero, so k=0.

 

[kuruman]

Your problem is that you do not understand the difference between a definite integral and an indefinite integral.
1. Definite integral
$$\int_{x_0}^x u~du=\frac{1}{2} \left. u^2 \right |_{x_0}^x=\frac{1}{2}x^2-\frac{1}{2}x_0^2.$$You just evaluate the antiderivative at the upper and lower limits and subtract the latter from the former.

2. Indefinite integral
$$\int u~du=\frac{1}{2} u^2 +k.$$There are no upper and lower limits; constant $k$ is a placeholder for the lower limit. In this particular problem you have a definite integral, you know the lower limit, therefore the placeholder is not needed or, as has already been shown, it must be zero.

 

[MichaelTam]

So, is my answer in the equation two is the simplest answer?
Why my answer in the equation 2 is still incorrect?

 

[MichaelTam]

Ops, I find my self have some typing error, the answer in the equation two is correct, thank you kuruman and haruspex!

 

[MichaelTam]

Are there any way to learn more calculus to get more strategy?

 

[haruspex]

Are there any way to learn more calculus to get more strategy?

It looks like you need to reread integration basics, paying attention to bounds and to definite and indefinite integrals.

 

[ThEmptyTree]

I got v(t) = -1/c ln( e^(-c v_0) + bc/m t ).

 

[haruspex]

I got v(t) = -1/c ln( e^(-c v_0) + bc/m t ).

Good.

 

[idekthanks]

Homework Statement:: MIT pretest.
Relevant Equations:: 𝐅⃗=−𝑏𝑒^(𝑐𝑣)𝐢̂ , find v(t), by using differential equation of F=maHello, would you mind sharing what course this is? I am familiar with edx but can’t find it

 

[haruspex]

@idekthanks , the thread is over a year old. @MichaelTam might not be visiting the forum any more.

 

[deepread]

@idekthanks Maybe this is a late reply, but I leave it for future readers.
The course contents are available here - https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/

 

[Delta2]

This problem becomes more complex if we alter the initial condition of $v_0$ to be not parallel to x-axis but to have some other direction. For example if $v_0$ is in the y-axis then the differential equation becomes $$m\frac{dv_x}{dt}=-be^{c\sqrt{v_0^2+v_x^2}}$$ and who can solve this