Integration by parts and approximation by power series

[Elvis 123456789]

Homework Statement

An object of mass m is initially at rest and is subject to a time-dependent force given by F = kte^(-λt), where k and λ are constants.
a) Find v(t) and x(t).
b) Show for small t that v = 1/2 *k/m t^2 and x = 1/6 *k/m t^3.
c) Find the object’s terminal velocity.

Homework Equations

The Attempt at a Solution

I showed my work for finding v(t) and getting the approximation for v(t) for small t. but I am missing the 1/2 at the front and I can't seem to find where it comes from.

 

[BvU]

Perhaps you need to further develop $e^{-\lambda t}$ ? The term linear in $t$ cancels, so you need to develop up to $t^2$

 

[Elvis 123456789]

Perhaps you need to further develop $e^{-\lambda t}$ ? The term ilinear in $t$ cancels, so you need to develop up to $t^2$

but if I include a t^2 term then I end up with a t^3 term that doesn't cancel

 

[BvU]

Yes. So the outcome will be correct up to order $t^2$. .

 

[BvU]

You can also do the development before the integration: $t\;e^{-\lambda t}= t -\lambda t^2$ and ignore higher orders

 

[Elvis 123456789]

You can also do the development before the integration: $t\;e^{-\lambda t}= t -\lambda t^2$ and ignore higher orders

Thanks for your help, i got the right answer. But can you help me understand something, why is it enough to include only up to the t^1 term before the integration, but after the integration we have to include the t^2 term?

 

[BvU]

"Integration increases order of $t$ by one" is the answer that comes to mind. But I agree with you that it makes a weird impression.