# SUVAT Equations ## Introduction to SUVAT - SUVAT equations describe the motion of objects experiencing **constant acceleration** - They are named after the five physical quantities they involve: - $s$ = displacement (m) - $u$ = initial velocity (m s$^{-1}$) - $v$ = final velocity (m s$^{-1}$) - $a$ = acceleration (m s$^{-2}$) - $t$ = time (s)

The SUVAT equations are mathematical relationships that connect displacement, initial velocity, final velocity, acceleration, and time for motion with constant acceleration in a straight line.

## The Four SUVAT Equations - There are four key equations, all provided on your data sheet: $$v = u + at$$ $$s = ut + \frac{1}{2}at^2$$ $$s = \frac{(v + u)}{2}t$$ $$v^2 = u^2 + 2as$$ - Each equation connects different combinations of the five variables - Choosing the right equation depends on which quantities you know and which you need to find ## Visualizing SUVAT Motion ### Velocity-Time Graphs - For constant acceleration, the velocity-time graph is always a **straight line** - The gradient of the line equals the acceleration - Example: An object starting at 5 m s$^{-1}$ with constant acceleration of 2 m s$^{-2}$

- This graph illustrates $v = u + at$ where $u = 5$ m s$^{-1}$ and $a = 2$ m s$^{-2}$ - The area under a velocity-time graph gives the displacement ### Displacement-Time Graphs - For constant acceleration, the displacement-time graph is a **curve** (parabolic if starting from origin) - The gradient at any point gives the instantaneous velocity - Example: Same object as above ($u = 5$ m s$^{-1}$, $a = 2$ m s$^{-2}$)

- This graph illustrates $s = ut + \frac{1}{2}at^2$ with the same values - The increasing steepness shows the object is speeding up ## Understanding the Variables ### Vector Quantities - All variables **except time** are vector quantities - This means they have both magnitude and direction - Direction is indicated by the sign (positive or negative) - You must maintain consistency with your chosen direction convention throughout calculations | Variable | Quantity | SI Unit | Vector? | |----------|----------|---------|---------| | $s$ | Displacement | m | Yes | | $u$ | Initial velocity | m s$^{-1}$ | Yes | | $v$ | Final velocity | m s$^{-1}$ | Yes | | $a$ | Acceleration | m s$^{-2}$ | Yes | | $t$ | Time | s | No | ### Common Scenarios to Recognize - **"Object begins from rest"** or if no initial velocity is mentioned → $u = 0$ - **"Comes to rest"** or **"stops"** → $v = 0$ - **"Falling freely under gravity"** (no air resistance) → $a = g = 9.81$ m s$^{-2}$

When choosing positive and negative directions, pick whichever makes sense for your problem - just ensure all vector quantities follow the same convention consistently.

### Direction Convention Example - If you define upward motion as positive for an object thrown vertically: - Upward displacement: $s$ is positive - Downward acceleration due to gravity: $a = -9.81$ m s$^{-2}$ (negative) - Initial upward velocity: $u$ is positive - Velocity at peak: $v = 0$ ## Applicability of SUVAT Equations - SUVAT equations **only work** when: - Acceleration is constant - Motion is in a straight line - Typical examples include: - A car braking with constant deceleration - An object falling through a uniform gravitational field (no air resistance) - A trolley rolling down a slope with constant acceleration ## Using SUVAT Equations: Systematic Approach ### Step 1: List Known and Unknown Variables - Write out all five SUVAT variables - Identify which are **given** in the question - Note which quantity you need to **find** - Use physical context to deduce implicit information - E.g., for free fall problems: $a = \pm 9.81$ m s$^{-2}$ - E.g., if something "launches from rest": $u = 0$ ### Step 2: Select the Appropriate Equation - Choose the equation containing: - The three or four variables you **know** - The one variable you **want to find** - Each equation omits one of the five variables - pick the one that omits the variable you neither know nor need | Equation | Missing Variable | |----------|-----------------| | $v = u + at$ | $s$ | | $s = ut + \frac{1}{2}at^2$ | $v$ | | $s = \frac{(v + u)}{2}t$ | $a$ | | $v^2 = u^2 + 2as$ | $t$ | ### Step 3: Substitute and Solve - **Convert all units to SI** before substituting (metres, seconds, etc.) - Insert the known values into your chosen equation - Rearrange algebraically to solve for the unknown - Include appropriate units in your final answer

Always check that all your quantities are in SI units before calculation - a common error is leaving distances in cm or km!

## Multi-Stage Problems - Some problems require using SUVAT equations **twice** in succession - The final value from the first stage becomes an initial value for the second stage - Examples include: - A rocket accelerating then coasting - Two connected masses where one stops before the other - A bouncing ball (separate calculations for upward and downward motion) ## Worked Example: Train Deceleration **Problem:** A train approaches a station at 50 m s$^{-1}$. The driver applies the brakes, causing uniform deceleration. The train must slow to 10 m s$^{-1}$ within 20 seconds to pass a speed restriction safely. Calculate the minimum distance the driver should be from the restriction point when braking begins. **Solution:** **Step 1: Identify known variables** - Initial velocity: $u = 50$ m s$^{-1}$ - Final velocity: $v = 10$ m s$^{-1}$ - Time taken: $t = 20$ s - Displacement: $s = ?$ (what we need to find) **Step 2: Choose equation** - We know $u$, $v$, and $t$ - We want to find $s$ - The equation linking these variables is: $$s = \frac{(v + u)}{2}t$$ **Step 3: Substitute and calculate** $$s = \frac{(10 + 50)}{2} \times 20$$ $$s = \frac{60}{2} \times 20$$ $$s = 30 \times 20 = 600 \text{ m}$$ The driver should begin braking at least **600 m** before the speed restriction point. ---

SUVAT problems are extremely common in exams - practice is the key to mastery! Work through as many different scenarios as possible.