Let's analyze the problem and solve the questions step by step. The graph shows the motion of a particle where the $x$-coordinate changes as a function of time. Below are the tasks requested:

Step-by-step Analysis:

1. Determine Velocity ($v_x$):

   • Velocity corresponds to the slope ($\frac{dx}{dt}$) of the $x(t)$ graph. Positive slope indicates positive velocity, and negative slope indicates negative velocity.

2. Determine Acceleration ($a_x$):

   • Acceleration corresponds to the change in the slope of the velocity ($\frac{dv_x}{dt}$). If the slope of the graph is increasing (convex curve), acceleration is positive. If it is decreasing (concave curve), acceleration is negative.

Answers to the questions:

(a) Järjestä pisteet P, Q, R, S niiden nopeuden ($v_x$) mukaan:

• Velocity ($v_x$) is given by the slope of the graph at each point:
  • $P$: Negative slope
  • $Q$: Zero slope
  • $R$: Positive slope
  • $S$: Zero slope
• Order: $P < Q = S < R$.

(b) Milloin $v_x$ on positiivinen, milloin yhtä suuri kuin nolla?:

• $v_x > 0$: Positive slope between $Q$ and $R$.
• $v_x = 0$: Zero slope at $Q$ and $S$.

(c) Järjestä pisteet niiden vauhdin mukaan ($|v_x|$):

• Speed ($|v_x|$) is the magnitude of the slope:
  • $P$: Steepest slope, highest $|v_x|$
  • $R$: Moderate slope, second-highest $|v_x|$
  • $Q = S$: Zero slope.
• Order: $P > R > Q = S$.

(d) Missä pisteissä kiihtyvyys on positiivinen?:

• Positive acceleration occurs where the graph is convex (slope is increasing):
  • Between $P$ and $Q$, and between $R$ and $S$.

(e) Missä pisteissä kiihtyvyys on negatiivinen?:

• Negative acceleration occurs where the graph is concave (slope is decreasing):
  • Between $Q$ and $R$.

(f) Missä pisteissä vauhti kasvaa?:

• Speed ($|v_x|$) increases when the graph's slope moves away from zero (either positively or negatively):
  • Between $P$ and $Q$, and between $R$ and $S$.

Would you like a more detailed explanation of any specific question or concept? Here are some related questions to expand your understanding:

1. How is the slope of a graph mathematically related to velocity?
2. What distinguishes velocity from speed in motion graphs?
3. How does the curvature of a graph relate to acceleration?
4. Why is acceleration positive when the slope increases, regardless of direction?
5. Can a particle have zero velocity but non-zero acceleration?

Tip: In problems with motion graphs, always focus on slopes for velocity and curvature for acceleration!