Equations of Motion

Velocity-Time Relation: $v = u + at$. This equation is used when displacement ($s$) is not known or required; it calculates the final velocity based on the rate of change over time.

Displacement-Time Relation: $s = ut + \frac{1}{2}at^2$. This is the primary tool for finding the position of an object when the final velocity ($v$) is unknown.

Average Velocity Relation: $s = \frac{(u + v)}{2}t$. This formula treats the motion as if the object traveled at a constant average velocity for the duration of the interval.

Time-Independent Relation: $v^2 = u^2 + 2as$. This equation is essential when the time interval ($t$) is unknown, relating the squares of the velocities to the displacement and acceleration.

Step 1: Define a Direction: Choose which direction is positive (e.g., upwards or to the right). All vectors pointing in the opposite direction must be treated as negative values in the calculation.

Step 2: List Knowns and Unknowns: Explicitly write down the values for $s, u, v, a,$ and $t$. Identify which variable you are trying to find and which variable is completely absent from the problem.

Step 3: Select the Equation: Choose the SUVAT equation that contains your knowns and the target unknown, but does not contain the variable that is missing from the problem context.

Step 4: Rearrange and Solve: Algebraically isolate the target variable before substituting numerical values to minimize calculation errors.

Identify Implicit Information: Look for keywords like 'starts from rest' ($u=0$), 'comes to a stop' ($v=0$), or 'dropped' ($u=0$ and $a = 9.81$ $m s^{-2}$ downward).

Check Units: Ensure all quantities are in standard SI units (meters, seconds, $m s^{-1}$, $m s^{-2}$) before substituting them into the formulas.

Sanity Check: Evaluate if the answer is physically reasonable. For example, if a car is braking, the final displacement should not be negative if the initial velocity was positive.

Sign Consistency: The most common mistake is mixing signs. If you define 'up' as positive, the acceleration due to gravity ($g$) must be entered as $-9.81$ $m s^{-2}$.

The 'Zero Acceleration' Trap: Students often try to use SUVAT when acceleration is changing. If a force is not constant, the acceleration is not constant, and these equations will yield incorrect results.

Squaring Negatives: In the formula $v^2 = u^2 + 2as$, remember that squaring a negative velocity results in a positive number. However, the $2as$ term can be negative if $a$ and $s$ have opposite signs (deceleration).

Time is Always Positive: If a quadratic rearrangement of $s = ut + \frac{1}{2}at^2$ gives two time values, usually only the positive one is physically meaningful, unless the negative one represents a time before the 'start' of the observed motion.