In the image, the problem involves calculating the speed of separation (denoted as $z$) between a car (mobil) and a train (kereta api), and determining the rates of change of the $x'$, $y'$, and $z'$ components over time.

Given:

• The car moves at a speed of $v_m = 30$ km/h.
• The train moves at a speed of $v_k = 20$ km/h.
• At $t = 0$, the car is positioned 0.5 km from the $y$-axis and the train is 1.5 km away along the $x$-axis.
• The car and the train are moving perpendicular to each other (the car moves along the $y$-axis and the train moves along the $x$-axis).

Approach:

Let’s define the following:

• $x(t)$ represents the position of the train along the $x$-axis, and $y(t)$ represents the position of the car along the $y$-axis.
• The distance $z(t)$ between the car and the train is the hypotenuse of a right triangle, with sides $x(t)$ and $y(t)$.

At any time $t$, the distance between the car and the train can be found using the Pythagorean theorem: $z(t) = \sqrt{x(t)^2 + y(t)^2}$

The rates $x'(t)$, $y'(t)$, and $z'(t)$ are the time derivatives of the positions and the distance, respectively. To find them, we can use:

1. Position Functions:

   • $x(t) = 1.5 - v_k t = 1.5 - 20t$ (since the train is moving towards the origin along the $x$-axis).
   • $y(t) = 0.5 + v_m t = 0.5 + 30t$ (since the car is moving away from the origin along the $y$-axis).

2. Velocity Components:

   • $x'(t) = -v_k = -20$ km/h (negative because the train is moving towards the origin).
   • $y'(t) = v_m = 30$ km/h.

3. Rate of Change of $z(t)$: Using the chain rule, we can compute $z'(t)$ as follows: $z'(t) = \frac{d}{dt} \left( \sqrt{x(t)^2 + y(t)^2} \right) = \frac{x(t) x'(t) + y(t) y'(t)}{z(t)}$ This gives the rate at which the distance between the car and the train changes.

Let me know if you'd like me to compute these values for a specific time $t$ or if you have any questions!

Expanding on this:

1. How do you find the initial distance $z(0)$ between the car and the train?
2. What would happen to $z(t)$ if the train or car speeds were different?
3. Can you derive the time when $z'(t) = 0$, meaning the separation distance stops changing momentarily?
4. How does the relationship between $x'(t)$, $y'(t)$, and $z'(t)$ depend on their directions of motion?
5. Can you calculate the angle of separation between the car and the train’s trajectory at any given time?

Tip:

In problems like this, breaking the motions into perpendicular components (like $x$ and $y$) helps simplify the calculation of distances and rates of change!