Gradient as Speed: The most fundamental principle of these graphs is that the gradient (slope) of the line represents the speed of the object. A steeper slope indicates a higher speed, while a shallower slope indicates a lower speed.
Mathematical Derivation: Since speed is defined as distance divided by time (), and the gradient of a line is the change in the y-value divided by the change in the x-value (), the gradient directly calculates the speed.
Linear vs. Non-Linear: A straight line indicates that the gradient is constant, meaning the object is moving at a uniform speed. Conversely, a curved line indicates a changing gradient, which signifies that the object is accelerating or decelerating.
| Graph Feature | Motion Represented | Speed Characteristic |
|---|---|---|
| Horizontal Line | Stationary | Speed is zero |
| Straight Diagonal | Constant Speed | Speed is uniform/unchanging |
| Upward Curve | Acceleration | Speed is increasing |
| Downward Curve | Deceleration | Speed is decreasing |
Unit Verification: Always check the units on both axes before performing calculations. If distance is in kilometers and time is in minutes, you must convert them to meters and seconds if the question asks for speed in .
Gradient Triangle Size: When calculating gradients, always draw the largest possible triangle that fits the data. Using a larger triangle reduces the percentage error in your measurements and leads to a more accurate speed calculation.
Tangent Precision: When drawing a tangent to a curve, ensure the line just touches the curve at the specific point. The angles between the curve and the tangent should appear equal on both sides of the point of contact.
Sanity Checks: If a graph shows a horizontal line, the speed calculation must result in zero. If your calculation for a flat line gives a value, you have likely confused the distance value with the change in distance.
