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Recently, I wrote an introductory physics exam where I encountered the following problem: We look at a bicyclist with the wind drag force $\vec{F_R}(t) = -k \vec{v}(t) |\vec{v}(t)|$ who accelerates with $F=50N$. The bicyclist is at rest initially, the wind is perpendicular to the direction of motion. The bicyclist moves only in one direction rectilinear.

I was pretty confused and couldn't answer the question. Especially the hint we got seemed strange to me: "Note that the wind is effectively coming from "skewed head on"". The other hint was that we should note that $\vec{v}(t)$ is the relative velocity, but that seemed obvious.

The exercise was to solve the equation of motion in two cases: wind speed >> speed of bicycle and wind speed << speed of bicycle. Also, we should determine the terminal velocity. Given was the wind speed as a scalar ($150 \frac{km}{h}$). Furthermore, m and k were given.

I would be perfectly fine if I was given either the wind vector or a scalar equation for the drag force, but I don't know what to do with this. Newtons 2nd law seemd a bit complicated in this form to me as it was clearly stated that the exam didn't involve nothing more advanced than a harmonic oscillator or maybe a parachute ODE. This is what I came up with: $$ F_{tot} = ma - k \vec{v}(t) |\vec{v}(t)| \\ terminal: \ F_{tot} = 0 \\ 0 = \frac{d^2 \vec{x}}{dt^2} -k \frac{d\vec{x}(t)}{dt} |\frac{\vec{x}(t)}{dt}| $$ I appreciate any help! The exercise is likely much easier than I imagine right now.

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Assume (explicitly) that "accelerates with 50N" means that the 50N force is applied in the direction of the line on which the body is constrained to move, so that you can express it as a vector and add it with the vector drag force. Split the drag force vector into vector components parallel to and perpendicular to the line on which the body is constrained to move. Sum the vectors and evaluate the equations of motion, remembering that the body is constrained to move only on the line parallel to the 50N force.

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  • $\begingroup$ So from what I understand, you want me to use $\vec{v}(t) = (v_0 + v_{wx})$ where $v_0$ is the velocity of the bicycle and $v_{wx}$ is the parallel component. But how does this help me? $v_{wx}$ is still unknown as I only have the scalar quantity and no exact angle at which the wind approaches the bicycle. $\endgroup$
    – arrhenius
    Commented Feb 10, 2023 at 14:24

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