The area under an acceleration-time graph gives the change in velocity
Motion & Kinematics - OCR A-Level Physics
Key rule
The area under an acceleration-time graph between two times equals the change in velocity over that interval. The gradient of the a-t graph is the rate of change of acceleration (called jerk; not assessed at A-level).
$$\Delta v = \int a \, dt = \text{area under } a\text{-}t \text{ graph}$$
Diagram pending
Three a-t graphs side by side. (a) Horizontal line at $a = 9.81 \text{ m s}^{-2}$ for free fall. (b) Decaying curve falling from $g$ to $0$ for an object reaching terminal velocity. (c) Rectangular pulse representing brief acceleration phase, with shaded area labelled $\Delta v$.
Will be replaced with a GeoGebra SVG in stream 2.
Case 1: Horizontal line, $a \neq 0$
Constant (uniform) acceleration. Area = $a \times t$, a rectangle. SUVAT applies. Free fall: $a = 9.81 \text{ m s}^{-2}$ (or $-9.81$ depending on sign convention).
Case 2: Line on the axis ($a = 0$)
Constant velocity. No change in $v$. Area under the line is zero.
Case 3: Decaying curve to zero
Object approaching terminal velocityThe constant velocity reached when the driving force on an object is exactly balanced by resistive forces, so the resultant force is zero.. Acceleration starts at $g$ and falls smoothly to zero as air resistance grows. SUVAT does not apply here.
- A car accelerates at $3 \text{ m s}^{-2}$ for $4 \text{ s}$, then coasts at $a = 0$ for $6 \text{ s}$, then decelerates at $-2 \text{ m s}^{-2}$ for $3 \text{ s}$. Change in velocity over the whole journey $= (3)(4) + (0)(6) + (-2)(3) = +6 \text{ m s}^{-1}$.
- For non-uniform $a$, use the trapezium rule or count squares to estimate the area.
- The a-t graph is the gradient of the v-t graph. Whenever the v-t graph is straight, the a-t graph is horizontal.
Common Mistake
LOW
Wrong: Reading the area under an a-t graph as displacement.
Right: Area under a-t gives change in velocity. Area under v-t gives displacement. The two graphs and their areas mean different things.
Right: Area under a-t gives change in velocity. Area under v-t gives displacement. The two graphs and their areas mean different things.
Examiner Tips and Tricks
- Students are often asked to sketch a-t graphs for falling objects.
- For free fallMotion under gravity alone, with no other forces acting. All objects in free fall near Earth's surface have the same acceleration, g = 9.81 m s⁻².: horizontal line at g.
- For falling with air resistanceThe opposition to currentThe rate of flow of chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C).. Measured in amperes (A). flow. The ratio of potential difference to currentThe rate of flow of chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C).. Measured in amperes (A).. Measured in ohms (Ω).: starts at g, curves down to zero as the object reaches terminal velocityThe constant velocity reached when the driving force on an object is exactly balanced by resistive forces, so the resultant force is zero..