How to find an object's velocity using only a graph
In short: If you have a graph showing an object's displacement over time, you are holding the key to its motion. By understanding the relationship between the slope of a line and its rate of change, you can use the power of differentiation to reveal exactly how fast something is moving.
To model motion along a straight line, we first establish a reference point and a positive direction. Displacement is then defined as the distance from this reference point, using plus or minus signs to indicate direction. When we plot this displacement against time, the resulting graph provides a visual map of the object's journey. To ensure accuracy, the units used for time and displacement must be consistent throughout the calculation. For example, one might use seconds for time and metres for displacement, which naturally leads to metres per second for velocity.
The critical insight lies in the gradient, or slope, of the graph. In a displacement-time graph, the gradient represents the rate of change of displacement with respect to time, which is the object's velocity. For instance, a straight line with a gradient of 6 km/h indicates a constant velocity of 6 km/h. Conversely, a curve where the gradient decreases over time suggests the velocity is continuously slowing down. A specific example of this is a man walking along a straight path where the horizontal axis represents hours and the vertical axis represents kilometres. In such a scenario, the gradient is measured in kilometres per hour.
This relationship is the foundation of differentiation. If you possess a mathematical formula for displacement in terms of time, you can apply differentiation to derive a specific formula for velocity. This process allows us to move from knowing where an object is to knowing how fast it is moving at any precise moment. While displacement is often denoted by the letter 's'—derived from the Latin 'spatium'—velocity is typically denoted by 'v' and time by 't'. It is worth noting that mathematicians often avoid using the letter 'x' for displacement to prevent confusion with the 'd' used in Leibniz notation. This precision is vital when performing the calculus required to find the derivative.
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Sources
- Summary by Intellect · source: Interpreting change in speed from velocity-time graph | Differential Calculus | Khan Academy
- Also: Introduction to differentiation: 3.1 Displacement and velocity | OpenLearn - Open University
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