Velocity-Time Graphs

• Gradient as Acceleration: The slope or gradient of a velocity-time graph at any point represents the instantaneous acceleration of the object. This is derived from the definition of acceleration: $a = \frac{\Delta v}{\Delta t}$

• Area as Displacement: The geometric area between the graph line and the time axis represents the change in displacement. This relationship exists because displacement is the integral of velocity with respect to time: $s = \int v \, dt$

• Linearity and Constant Motion: A straight line on a v-t graph indicates constant acceleration. If the line is horizontal, the acceleration is zero, meaning the object is moving at a constant velocity.

Calculating Acceleration

• To find the acceleration during a specific interval, identify two points $(t_1, v_1)$ and $(t_2, v_2)$ on a straight section of the graph. Calculate the gradient using the formula: $a = \frac{v_2 - v_1}{t_2 - t_1}$

Calculating Displacement and Distance

• Total Displacement: Calculate the area of all regions between the graph and the x-axis. Areas above the x-axis are treated as positive, while areas below the x-axis are treated as negative. Sum them algebraically.
• Total Distance: Calculate the absolute area of all regions (treating all areas as positive) and sum them together. Distance is a scalar and does not account for direction.
• Geometric Shapes: Most exam problems involve composite shapes like triangles, rectangles, and trapeziums. Use standard area formulas: $Area_{triangle} = \frac{1}{2}bh$ and $Area_{trapezium} = \frac{1}{2}(a+b)h$.

• It is vital to distinguish between velocity-time and displacement-time graphs, as the same visual features represent different physical properties.

• Check the Units: Always verify the units on both axes. If time is in minutes or hours, you may need to convert to seconds to ensure consistency with velocity in $m s^{-1}$.

• Identify 'At Rest': An object is instantaneously at rest whenever the graph crosses or touches the x-axis ($v = 0$). Do not confuse this with a horizontal line, which simply means constant velocity.

• Sign Conventions: Be extra vigilant with negative gradients. A negative gradient above the x-axis means the object is slowing down (decelerating) while moving forward. A negative gradient below the x-axis means the object is speeding up while moving backward.

• Sanity Check: If a question asks for distance, your answer must be positive. If it asks for displacement, the answer can be zero or negative if the object returns to or passes the starting point.

• Confusing Distance and Displacement: Students often forget that areas below the x-axis must be subtracted for displacement but added for distance. This leads to incorrect results in 'round-trip' scenarios.

• Misinterpreting the Origin: Touching the x-axis on a v-t graph means the object has stopped, NOT that it has returned to the starting position. Returning to the starting position is indicated when the net area (above minus below) equals zero.

• Gradient Errors: Ensure you use the change in velocity over the change in time, not just the velocity value at a single point, when calculating acceleration.