How Do You Find Velocity or Acceleration From a Graph?

Velocity is the gradient of a position versus time graph, while acceleration is the gradient of a velocity versus time graph. For instantaneous values on curved graphs, calculate the gradient of a tangent at the required time.

The Reasoning and Method

In IB Physics A.1 Kinematics (SL and HL), velocity is the rate of change of position and acceleration is the rate of change of velocity:

$$v=\frac{\Delta x}{\Delta t}, \qquad a=\frac{\Delta v}{\Delta t}$$

These quotients give an average velocity or average acceleration over a time interval. At one instant, use the derivative: $v=\frac{dx}{dt}$ and $a=\frac{dv}{dt}$. Graphically, this is the gradient of the tangent. A straight section has a constant gradient, so the corresponding velocity or acceleration is constant.

Therefore, the graph's vertical quantity determines what its gradient represents.

Check each axis scale before taking readings. A horizontal position versus time section means the object is stationary, whereas a horizontal velocity versus time section means constant velocity and zero acceleration.

For example, if position changes from $4.0\,\text{m}$ at $2.0\,\text{s}$ to $16.0\,\text{m}$ at $5.0\,\text{s}$, then

$$v=\frac{16.0-4.0}{5.0-2.0}=4.0\,\text{m s}^{-1}.$$

If velocity changes from $-2.0\,\text{m s}^{-1}$ to $6.0\,\text{m s}^{-1}$ over $4.0\,\text{s}$, then

$$a=\frac{6.0-(-2.0)}{4.0}=2.0\,\text{m s}^{-2}.$$

A negative gradient means negative velocity on an $x$-$t$ graph or negative acceleration on a $v$-$t$ graph. The common misconception is that negative acceleration always means slowing down. It does not: an object speeds up when velocity and acceleration have the same sign and slows down when their signs are opposite.

IB Exam Technique

For a curved graph, draw a clear tangent at the specified point and use a large gradient triangle, preferably covering more than half the tangent. Show both coordinate readings, retain signs, and include units. Do not confuse gradient with area: the area under a velocity versus time graph gives displacement, not acceleration.