Final answer:
To write v(r) as dr/dt, differentiate the equation for discharging a capacitor through a resistor. Rearrange the equation and integrate to find r(t). The behavior of r(t) depends on the specific values of the variables in the equation.
Step-by-step explanation:
The question asks to write v(r) as dr/dt and rearrange the previous result to find r(t) by integrating it. This question seems to be related to a topic in physics, specifically in the context of discharging a capacitor through a resistor.
To write v(r) as dr/dt, we can differentiate the equation that relates voltage and time for discharging a capacitor through a resistor. The equation is: V = Vo * exp(-t / RC), where V is the voltage at time t, Vo is the initial voltage, R is the resistance, and C is the capacitance.
By differentiating this equation with respect to time (t), we get: dV/dt = (-Vo / RC) * exp(-t / RC). Rearranging this equation, we have: dV = (-Vo / RC) * exp(-t / RC) * dt. Now, we substitute dV with dr (representing the change in the distance r): dr = (-Vo / RC) * exp(-t / RC) * dt.
To find r(t), we can integrate both sides of the equation: ∫dr = ∫(-Vo / RC) * exp(-t / RC) * dt.
Integrating the left side of the equation will give us r(t) + C1, where C1 is the constant of integration. And for the right side, we use the substitution method to integrate the exponential term.
Finally, after integrating and solving for r(t), we may determine whether it increases exponentially, decreases exponentially, remains constant, or follows a sinusoidal pattern based on the values of Vo, R, and C.