Let's analyze the problem from the image and address each part.

The task revolves around interpreting a motion graph, $x(t)$, showing the x-coordinate as a function of time. Points $P, Q, R, S$ are marked on the graph, and the questions are related to velocity, acceleration, and their signs.

Definitions:

1. Velocity ($v_x$): The slope of the tangent to the curve at a point.

   • Positive slope → $v_x > 0$
   • Negative slope → $v_x < 0$
   • Horizontal tangent → $v_x = 0$

2. Acceleration: The rate of change of $v_x$ (slope of the $v_x$ vs. $t$ graph).

   • $a > 0$: Slope of $v_x$ increasing
   • $a < 0$: Slope of $v_x$ decreasing

(a) Arrange points $P, Q, R, S$ based on velocity $v_x$:

• Point $P$: $v_x < 0$ (negative slope).
• Point $Q$: $v_x = 0$ (flat slope).
• Point $R$: $v_x > 0$ (positive slope).
• Point $S$: $v_x = 0$ (flat slope).

Order (ascending velocity): $P, Q, S, R$.

(b) When is $v_x > 0$, $v_x = 0$, $v_x < 0$?

• $v_x > 0$: Between $Q$ and $R$ (upward slope).
• $v_x = 0$: At $Q$ and $S$ (horizontal tangent).
• $v_x < 0$: Before $Q$ (downward slope) and after $S$.

(c) Arrange points based on speed (magnitude of $v_x$):

• Speed is the absolute value of velocity:
  • $P$: Large negative slope → High speed.
  • $Q$: $v_x = 0$ → Zero speed.
  • $R$: Positive slope → High speed.
  • $S$: $v_x = 0$ → Zero speed.

Order (ascending speed): $Q, S, R, P$.

(d) At which points is acceleration positive?

• Positive acceleration ($a > 0$): $v_x$ increases.
  • Between $P$ and $Q$ and between $S$ and $R$.

(e) At which points is acceleration negative?

• Negative acceleration ($a < 0$): $v_x$ decreases.
  • Between $Q$ and $S$.

(f) Where does the speed increase?

• Speed increases if $v_x$ moves away from zero (positive or negative acceleration and velocity alignment):
  • Between $P$ and $Q$, and between $R$ and $S$.

Would you like detailed calculations or further clarifications?

Related Questions:

1. How do we calculate acceleration from such graphs?
2. How is the concept of slope linked to instantaneous velocity?
3. How do we identify inflection points for acceleration?
4. What is the significance of horizontal tangents in motion graphs?
5. How does curvature affect interpretation in such problems?

Tip: Always check the slope direction for velocity and its rate of change for acceleration in motion graphs.