Gradients in Kinematic Graphs

Worked Example: Linear Graph

A car travelling on a straight road increases its velocity from $12\text{ m/s}$ to $28\text{ m/s}$. This linear change occurs between $t = 3\text{ s}$ and $t = 7\text{ s}$ on a velocity-time graph. Calculate the acceleration of the car.

Step 1: State the formula for acceleration.

• $\text{Acceleration} = \frac{\text{Change in Velocity}}{\text{Time Taken}} = \frac{v - u}{t}$

Step 2: Identify the coordinates $(3, 12)$ and $(7, 28)$ and substitute them into the gradient formula.

• $\text{Change in } y \text{ (Velocity)} = 28 - 12 = 16\text{ m/s}$

• $\text{Change in } x \text{ (Time)} = 7 - 3 = 4\text{ s}$

Step 3: Calculate the final answer and include appropriate units.

• $\text{Acceleration} = \frac{16}{4} = 4\text{ m/s}^2$

Estimating Gradients on Curved Graphs

If a graph features a curved line, it means the rate of change is not constant. An increasing curve on a distance-time graph means the object is speeding up, while a decreasing gradient means it is slowing down.

• To find the instantaneous rate of change (the exact speed or acceleration at a single moment in time), you must draw a tangent.
• A tangent is a straight line drawn by eye using a ruler that touches the curve at exactly one point without crossing it.
• Alternatively, drawing a chord (a straight line connecting two distinct points on the curve) will give you the average rate of change over that time interval.

Worked Example: Curved Graph

A distance-time graph shows the curved path of an accelerating cyclist. Estimate the instantaneous speed of the cyclist at exactly $t = 5\text{ s}$.

Step 1: Draw a tangent to the curve by eye at the requested time.

• Place a ruler so it perfectly grazes the curve at exactly $t = 5\text{ s}$.

Step 2: Pick two points on the tangent line that are far apart.

• Let's assume our drawn tangent passes clearly through the coordinates $(2, 10)$ and $(8, 46)$.

Step 3: Substitute these values into the gradient formula.

• $\text{Change in } y \text{ (Distance)} = 46 - 10 = 36\text{ m}$

• $\text{Change in } x \text{ (Time)} = 8 - 2 = 6\text{ s}$

Step 4: Calculate the final estimate with appropriate units.

• $\text{Speed} = \frac{36}{6} = 6\text{ m/s}$

Units and Axis Interpretation

Always double-check the labels on the $y$-axis before interpreting a graph, as flat lines mean completely different things depending on the context. A horizontal line (zero gradient) on a distance-time graph means the object is stationary. However, a horizontal line on a velocity-time graph does not mean the object has stopped; it means it is moving at a constant velocity with zero acceleration.

• You can always derive the correct units for your answer by dividing the $y$-axis unit by the $x$-axis unit.
• For example, dividing metres ($m$) by seconds ($s$) gives $m/s$ for velocity.
• Dividing velocity ($m/s$) by seconds ($s$) gives $m/s^2$ for acceleration.
• If an exam question asks for the overall average speed of an entire journey, do not use gradients. Instead, divide the total distance travelled by the total time taken.