Question 3 · Matemàtiques URV Exemples 2025

Question 3 (3 pts). Vertical motion with aerodynamic drag

An object of mass $m$ travels vertically through the atmosphere. The forces acting on the body are solely its weight and aerodynamic air resistance, which is directly proportional to its velocity (drag constant $k$). Consequently, the differential equation governing the velocity as a function of time is:$$m\frac{dv}{dt} = -mg - kv$$where $g = 9.8\text{ m/s}^2$. Prior observations establish that in the long run the object tends to fall at a constant terminal speed of $10\text{ m/s}$.

a)

Find the numerical value of the ratio $m/k$ for this object.

b)

Assuming the object is launched vertically upwards with an initial velocity $v(0) = v_0$, find an expression in terms of $v_0$ for the time during which the object continues to ascend.

c)

Assuming the initial height at $t = 0$ is zero ($x(0) = 0$), determine the height $x(t)$ as a function of time and the parameter $v_0$, and compute the maximum height attained by the object.