Position-Time Graphs

A position–time ($x$–$t$) graph is a visual record of an object's motion. Its slope tells you velocity; its curvature tells you about acceleration.

Concept

For a graph with time on the horizontal axis and position on the vertical:

Mathematical Form

If $x = x_0 + ut + \tfrac{1}{2}at^2$:

• Linear graph ($a = 0$): $x = x_0 + ut$.
• Parabolic graph: opens up for $a > 0$, opens down for $a < 0$.

Average and Instantaneous Velocity

• Average velocity between $t_1$ and $t_2$: slope of the secant connecting the two points.
• Instantaneous velocity at $t$: slope of the tangent at that point.

Sketching Tips

1. Mark axes with sensible scales.
2. Plot known points first; then connect using the appropriate shape (line, parabola).
3. The point where $x$ is maximum or minimum corresponds to $v = 0$ (slope = 0).

Worked Example

Q: An object's $x$-$t$ graph is given by $x = 2 + 4t - t^2$ m (with $t$ in s). Sketch (mentally) and find: (a) velocity at $t = 1$ s, (b) when does the object momentarily stop, (c) maximum position.

Solution:

(a) $v = dx/dt = 4 - 2t$; at $t = 1$, $v = 2$ m/s.

(b) Stops when $v = 0$: $4 - 2t = 0 \Rightarrow t = 2$ s.

(c) Maximum $x$ at $t = 2$: $x_{\max} = 2 + 8 - 4 = 6$ m.

The graph is a downward parabola opening down, peaking at $(2, 6)$, with $x_0 = 2$ at $t = 0$.

Common Confusions

• A graph that goes "up" doesn't mean the object goes "up" — only that position increases (could be eastward).
• A steep slope means high speed; a gentle slope means low speed; a horizontal segment means rest.
• A curve's curvature (not slope) signals acceleration. Constant slope = no acceleration even if slope is large.

Key Takeaways

• Slope of $x$–$t$ = velocity.
• Curvature of $x$–$t$ = sign of acceleration.
• Constant velocity → straight line; constant acceleration → parabola.
• Velocity is zero at points where $x$-$t$ has zero slope (turning points).