Velocity-Time Graphs

A velocity-time ($v$-$t$) graph is one of the most useful diagnostic tools in kinematics. Its slope is the acceleration; the area beneath it is the displacement.

Concept

Why Area = Displacement

$x(t) = x_0 + \int_0^t v\,dt'.$ The integral is the geometric area between the curve and the $t$-axis.

Sign Conventions

• $v > 0$: motion in chosen positive direction.
• $v < 0$: motion in negative direction.
• Net displacement = (positive area) − (negative area).

Velocity vs. Speed Graphs

A $v$-$t$ graph shows signed velocity. A $|v|$-$t$ (speed) graph only shows magnitude — slope of $|v|$ doesn't equal $|a|$ when sign changes.

Worked Example

Q: A car's $v$-$t$ graph:

• $0 \le t \le 5$ s: $v = 4t$ (accelerates uniformly from rest to 20 m/s).
• $5 \le t \le 10$ s: $v = 20$ m/s (uniform).
• $10 \le t \le 15$ s: $v$ decreases linearly to 0.

Find (a) acceleration in each phase, (b) total distance.

Solution:

Phase 1: $a_1 = 20/5 = 4$ m/s².

Phase 2: $a_2 = 0$.

Phase 3: $a_3 = (0 - 20)/(15-10) = -4$ m/s².

Distances (areas under graph):

• Phase 1: $\tfrac{1}{2}(5)(20) = 50$ m.
• Phase 2: $5 \times 20 = 100$ m.
• Phase 3: $\tfrac{1}{2}(5)(20) = 50$ m.

Total: $200$ m.

Reading Multiple Features

• Speeding up or slowing down? Sign of slope vs sign of velocity. Same sign → speeding up; opposite → slowing down.
• Stationary? $v$ on the $t$-axis (i.e. $v = 0$).
• Direction reversal? $v$ crosses the $t$-axis.

Common Confusions

• "Area on a $v$-$t$ graph gives distance." — It gives displacement (which is signed). For distance, take absolute values of each piece.
• "If $v$ increases, the object speeds up." — Only if $v > 0$; if $v < 0$, "increasing" (less negative) actually means slowing down.
• "Steeper slope = faster." — Steeper slope on $v$-$t$ = greater acceleration, not necessarily greater speed.

Key Takeaways

• Slope of $v$-$t$ = acceleration.
• Area under $v$-$t$ = displacement.
• Distance = sum of |areas| of each piece.
• Direction reverses where $v$ crosses zero.
• Position-time and velocity-time graphs are complementary diagnostic tools.