Equations of Motion

• The Definition of Acceleration: Acceleration is the rate of change of velocity, expressed as $a = \frac{v - u}{t}$. Rearranging this fundamental definition yields the first equation: $v = u + at$.

• Average Velocity: For an object with constant acceleration, the average velocity is simply the arithmetic mean of the initial and final velocities: $v_{avg} = \frac{u + v}{2}$.

• Geometric Interpretation: On a velocity-time graph, the slope of the line represents the acceleration, while the area under the graph represents the total displacement traveled.

• Displacement Derivation: By multiplying average velocity by time, we find the displacement: $x = \left(\frac{u + v}{2}\right)t$. Substituting $v = u + at$ into this leads to the quadratic form $x = ut + \frac{1}{2}at^2$.

The SUVAT Method

• Step 1: Inventory Variables: List the five variables ($u, v, a, x, t$) and identify which three are known and which one is the target unknown.
• Step 2: Select the Equation: Choose the equation that contains your three knowns and your one unknown, effectively 'ignoring' the fifth variable.
• Step 3: Sign Convention: Establish a positive direction (e.g., upwards is positive). If an object is falling, its acceleration $g$ and downward displacement will be negative.
• Step 4: Solve and Verify: Substitute values into the chosen formula and solve. Always check if the magnitude and sign of the result make physical sense (e.g., time cannot be negative).

• Distance vs. Displacement: In these equations, $x$ represents displacement (the straight-line change in position), not necessarily the total distance traveled if the object changes direction.
• Deceleration: This is simply acceleration in the opposite direction of motion. In calculations, deceleration is treated as a negative acceleration value if the initial velocity is positive.

• Identify Hidden Values: Look for keywords like 'starts from rest' ($u = 0$), 'comes to a stop' ($v = 0$), or 'dropped' ($a = g \approx 10 \text{ m/s}^2$).

• Unit Consistency: Always check for units like $km/h$ or minutes. Convert everything to meters and seconds before starting the calculation to avoid massive power-of-ten errors.

• The 'No-Time' Equation: If the problem does not provide or ask for time, immediately look to use $v^2 = u^2 + 2ax$. This is the most common shortcut in complex exam problems.

• Sanity Checks: If you calculate a car's final velocity to be $5000 \text{ m/s}$, re-check your algebra or unit conversions; the answer should be physically realistic.

• Non-Constant Acceleration: A common error is applying these equations to circular motion or varying forces (like air resistance). These formulas fail if acceleration changes.

• Squaring Errors: In the formula $v^2 = u^2 + 2ax$, students often forget to square the initial velocity or fail to take the square root at the end to find $v$.

• Directional Confusion: Forgetting that $u$ and $a$ can have opposite signs (e.g., an object thrown up while gravity pulls down) is the leading cause of incorrect displacement results.