SUVAT in 1D

Constant Acceleration: The fundamental requirement for using SUVAT equations is that the acceleration must remain uniform throughout the period of motion being analyzed.

Graphical Foundation: The equations are derived from a Velocity-Time (v-t) graph, where a straight line represents constant acceleration; the gradient of this line equals the acceleration ($a$).

Area as Displacement: The area under a velocity-time graph represents the total displacement ($s$) of the object, which forms the geometric basis for the displacement formulas.

Calculus Connection: These formulas can also be derived by integrating constant acceleration with respect to time to find velocity, and then integrating velocity to find displacement.

Equation 1 (No s): $v = u + at$. This relates velocity and time, derived directly from the definition of acceleration as the rate of change of velocity.

Equation 2 (No a): $s = \frac{1}{2}(u + v)t$. This calculates displacement using the average velocity of the object over the time interval.

Equation 3 (No v): $s = ut + \frac{1}{2}at^2$. This is the most common formula for finding displacement when the final velocity is unknown.

Equation 4 (No u): $s = vt - \frac{1}{2}at^2$. This is a variation used when the initial velocity is unknown but the final state is defined.

Equation 5 (No t): $v^2 = u^2 + 2as$. This is the only equation that does not require time, making it essential for problems where the duration of motion is not provided.

Step 1: Define Direction: Explicitly choose a positive direction (e.g., upwards or to the right) and stick to it for all vector values in the problem.

Step 2: List Knowns: Write down the values for the three variables provided in the text, paying close attention to keywords like 'at rest' ($u=0$) or 'stops' ($v=0$).

Step 3: Identify the Goal: Determine which variable you need to find and which variable is completely irrelevant to the specific question.

Step 4: Select Equation: Choose the SUVAT equation that contains your three knowns and your one target variable, but excludes the irrelevant variable.

Step 5: Solve and Verify: Substitute the values into the equation, solve for the unknown, and check if the magnitude and sign of the result make physical sense.

Deceleration vs. Negative Acceleration: Deceleration specifically refers to slowing down (velocity and acceleration have opposite signs), whereas negative acceleration simply means acceleration in the negative direction.

Initial vs. Final States: In multi-stage problems, the final velocity ($v$) of the first stage becomes the initial velocity ($u$) for the second stage.

Check for Constant Acceleration: Always verify that acceleration is constant before applying SUVAT; if acceleration changes (e.g., a car braking then accelerating), split the problem into two distinct stages.

Sign Consistency: A common mistake is mixing signs; if 'up' is positive, then gravity ($g$) must be entered as $-9.8$ or $-10$ depending on the exam board's specification.

Hidden Zeros: Look for 'starts from rest' ($u=0$), 'comes to a halt' ($v=0$), or 'returns to the origin' ($s=0$) to find the three necessary variables.

Unit Conversion: Ensure all units are converted to the standard $m$, $s$, and $m/s$ before calculation; for example, convert $km/h$ to $m/s$ by dividing by $3.6$.