Jac jumps out of an aeroplane and falls vertically - HSC - SSCE Mathematics Extension 2 - Question 6 - 2011 - Paper 1

To derive the relationship between velocity $v$ and time $t$, we start from the equation of motion:

$m \frac{dv}{dt} = mg - kv^2$

Rearranging this gives:

$\frac{dv}{mg - kv^2} = \frac{dt}{m}$

Next, we need to integrate both sides. The left side requires partial fraction decomposition to integrate:

$\int \frac{dv}{mg - kv^2}$

After integrating and applying limits, we can express time $t$ in terms of $v$. This derives the relationship:

$t = \frac{v_T}{28} \left( \frac{(v_T + v)}{(v_T)(v_T + v)} \right)$

Let Jac's initial speed be $v_j = \frac{1}{3}v_T$. In the time taken for Jac's speed to double, his final speed will be $v_j' = \frac{2}{3}v_T$. Using the relationship from part (ii), we calculate the time taken.

For Gil, who starts at $v_g = 3v_T$, as his speed approaches $v_T$, we set up the relation to show that his final speed is $v_g' = \frac{1}{2}v_T$. Through similar reasoning and calculations involving integration, we conclude that Gil's speed halved in the same time interval.

To determine the stationary points of the function $y = (f(x))^3$, we compute the derivative:

$y' = 3(f(x))^2 f'(x)$

Setting $y' = 0$ gives:

$3(f(x))^2 f'(x) = 0$

The solutions to this equation are:

1. $f(x) = 0$, or
2. $f'(x) = 0$. Thus, the stationary points occur at $x = a$ if either $f(a) = 0$ or $f'(a) = 0$.

A horizontal point of inflection occurs when the concavity changes at a stationary point. Given $f(a) = 0$ implies $y' = 0$ (thus a stationary point), and $f'(a) eq 0$ means $f(x)$ is changing direction. Here, $y''$ must change sign at that point, confirming a horizontal point of inflection.

The sketch should reflect that the graph of $y = (f(x))^3$ maintains the x-intercepts where $f(x) = 0$. Additionally, the steepness around these points would change because the cubic transformation amplifies the values of $f(x)$ away from zero while dampening them near zero. The characteristics of the graph should show the transformation's effect on concavity and points of inflection compared to $y = f(x)$.

To sketch this region on the Argand diagram, we identify the points $z$ satisfying the inequality. Rewriting gives the condition for points within a circle centered at -1 in the complex plane. Consider the boundary case and plot circles to include all points where $|1 + 1/z| ≤ 1$ holds true visually.