Gradient of a Displacement-Time Graph

• The mathematical foundation of the gradient is the ratio of the vertical change to the horizontal change, expressed as $v = \frac{\Delta s}{\Delta t}$.

• For a linear graph, the velocity is constant because the ratio of displacement to time remains uniform at every point along the line.

• In calculus terms, the velocity at any specific instant is the first derivative of the displacement function with respect to time: $v = \frac{ds}{dt}$.

• The steepness of the slope is directly proportional to the magnitude of the velocity; a steeper line indicates a higher speed, while a shallower line indicates a lower speed.

Calculating Constant Velocity

• Identify two distinct points on a straight-line segment: $(t_1, s_1)$ and $(t_2, s_2)$.
• Apply the gradient formula: $v = \frac{s_2 - s_1}{t_2 - t_1}$
• Ensure that the units are consistent, typically resulting in meters per second ($m/s$ or $ms^{-1}$).

Analyzing Non-Linear Motion

• If the graph is curved, the velocity is changing, which implies Acceleration or Deceleration.
• To find the Instantaneous Velocity at a specific time, draw a Tangent to the curve at that point and calculate the gradient of that tangent line.
• To find the Average Velocity over a time interval, draw a Chord (a straight line connecting two points on the curve) and calculate its gradient.

• Check the Y-Axis: Always verify if the vertical axis is 'Displacement' or 'Distance'. A downward slope on a displacement graph is valid, but on a distance graph, it usually indicates a physical impossibility or a misunderstanding of the variables.

• Sign Convention: In exams, a negative gradient often means the object is moving back toward the starting point. Don't drop the negative sign unless the question specifically asks for 'speed'.

• Tangent Precision: When drawing a tangent to a curve, ensure the line just 'touches' the point and has an equal amount of space between the curve and the line on both sides of the point to maximize accuracy.

• Units Check: Always look at the units on the axes. If displacement is in kilometers and time is in minutes, your gradient will be in $km/min$, which may need conversion to $m/s$.

• Confusing Gradient with Area: Students often mistakenly try to calculate the area under a displacement-time graph. In kinematics, the area under a displacement-time graph has no standard physical meaning; only the gradient (velocity) is useful.

• Misinterpreting 'Downwards': A line sloping downwards does not necessarily mean the object is slowing down; it means the object is moving in the negative direction. An object can have a very steep negative gradient, meaning it is moving very fast in reverse.

• Curvature Confusion: A curve that gets steeper (increasing gradient) represents acceleration, while a curve that flattens out (decreasing gradient) represents deceleration.