Deriving the suvat Formulas

• Acceleration as Gradient: On a velocity-time graph, acceleration is defined as the rate of change of velocity. For constant acceleration, the graph is a straight line where the gradient $a = \frac{v - u}{t}$, which rearranges to the first formula: $v = u + at$.

• Displacement as Area: The total displacement is the area under the velocity-time graph. For a straight line, this area forms a trapezium with parallel sides of lengths $u$ and $v$ and a height of $t$.

• Trapezium Rule: Calculating the area of this trapezium gives the formula $s = \frac{1}{2}(u + v)t$, which represents the displacement as the average velocity multiplied by time.

• Geometric Decomposition: By splitting the area into a rectangle (area $ut$) and a triangle (area $\frac{1}{2}at^2$), we derive $s = ut + \frac{1}{2}at^2$. Alternatively, using the rectangle $vt$ and subtracting the upper triangle gives $s = vt - \frac{1}{2}at^2$.

• Integrating Acceleration: Since acceleration $a$ is the derivative of velocity with respect to time, velocity is found by integrating acceleration: $v = \int a \, dt = at + C$. Using the initial condition that $v = u$ when $t = 0$, we find $C = u$, resulting in $v = u + at$.

• Integrating Velocity: Displacement is the integral of velocity over time. Substituting the expression for $v$, we get $s = \int (u + at) \, dt = ut + \frac{1}{2}at^2 + C$. Setting $s = 0$ at $t = 0$ yields $C = 0$, giving $s = ut + \frac{1}{2}at^2$.

• Boundary Conditions: Calculus derivations rely on defining specific limits or constants of integration based on the starting state of the system, typically at $t=0$.

• Displacement vs. Distance: In suvat derivations, $s$ refers to the change in position from the start, not the total path length. If an object reverses direction, displacement may be zero while distance is large.

• Algebraic vs. Geometric: While geometric derivations are intuitive for constant acceleration, calculus derivations are more robust as they connect kinematics to the broader mathematical laws of motion.

Comparison of Derivation Bases

• Identify the Missing Variable: Each suvat equation omits exactly one of the five variables. When deriving or applying them, identify which variable is not mentioned to select the correct starting point.

• Check for Uniformity: Always verify that the problem states 'constant acceleration' or 'uniform motion' before assuming suvat derivations apply.

• Sign Consistency: When deriving $v^2 = u^2 + 2as$, ensure that the signs for $a$ and $s$ are consistent with the chosen positive direction to avoid imaginary numbers or incorrect magnitudes.

• Dimensional Analysis: Verify derivations by checking units. For example, in $s = ut + \frac{1}{2}at^2$, both terms on the right must have units of length ($m/s \cdot s = m$ and $m/s^2 \cdot s^2 = m$).