No problem, glad I could help!

FeDeX_LaTeX · May 5, 2014

Kinematics -- Vector DE

Homework Statement

Printed homework problem 1 on a particle of mass m under gravity and air resistance proportional to the cube of its speed, with parts (a), (b) and (c) asking for the governing vector equation, the velocity with gravity neglected, and the velocity and position with air resistance neglected.

The Attempt at a Solution

Here's their solution.

Handwritten solution with parts (a) and (b): N2L gives sum F = d/dt p = m dv/dt, leading to m dv/dt = -mg z-hat - a(v dot v)v; part (b) manipulates v dot m dv/dt = -a(v dot v)^2 through to v = u(1 + 2av^2 t/m)^(-1/2).

I'm stuck on (b). The first and second lines are fine. But how do they get from the second line to the third line? And then in the fourth line, why are they integrating from u^2 to v^2?

EDIT: Sorry for the image size.

Chestermiller · May 5, 2014

They make the substitution v² = x.

Chet

FeDeX_LaTeX · May 5, 2014

Chestermiller said:

They make the substitution v² = x.

Chet

That's it?! I thought x denoted position (as a function of time)! Was losing my mind over this. Thanks.

No problem, glad I could help!

Homework Statement

The Attempt at a Solution

Discussion

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