This section explores the fundamental formulae known as SUVAT equations, which are utilized for solving problems involving uniformly accelerated motion in a straight line. These equations are crucial for understanding the kinematics of various objects, ranging from cars in motion to objects in free fall, under constant acceleration.

## Understanding SUVAT Equations

SUVAT equations relate 5 key elements: displacement $s$, initial velocity $u$, final velocity $v$, acceleration $a$, and time $t$. They're used when acceleration is constant.

**The Five Equations:**

**1. Final Velocity: **$v = u + at$

- Finds final velocity from initial velocity, acceleration, and time.

**2. Displacement 1: **$s = ut + \frac{1}{2}at^2$

- Calculates displacement using initial velocity, acceleration, and time.

**3. Displacement 2: **$s = \frac{u+v}{2} \cdot t$

- Another way to find displacement, using average velocity and time.

**4. Velocity-Squared: **$v^2 = u^2 + 2as$

- Connects velocity squares with displacement and acceleration.

**5. Displacement 3: **$s = vt - \frac{1}{2}at^2$

- A displacement formula using final velocity and time.

## Application in Problem-Solving

**Example 1: Finding Final Velocity**

**Question: **A car goes from rest to $3 \, \text{m/s}^2$ acceleration for $4$ seconds. What's its final velocity?

**Solution:**

**1. Known Values:**

- Initial velocity $(u): 0 \, \text{m/s}$ (rest).
- Acceleration $(a): 3 \, \text{m/s}^2$.
- Time ((t)): $4$ seconds.

**2. Formula: **

- $v = u + at$.

**3. Substitute:**

- $v = 0 + 3 \times 4$.

**4. Calculate:**

- $v = 12 \, \text{m/s}$.

**5. Answer:**

- Final velocity: $12 \, \text{m/s}$.

### Example 2: Calculating Displacement

**Question: **How far does the car move in those 4 seconds?

**Solution:**

**1. Known Values:**

- Initial velocity $(u): 0 \, \text{m/s}$ (rest).
- Acceleration $(a): 3 \, \text{m/s}^2$.
- Time $(t)$: $4$ seconds.

**2. Formula: **

- $s = ut + \frac{1}{2}at^2$.

**3. Substitute:**

- $s = 0 \times 4 + \frac{1}{2} \times 3 \times 4^2$.

**4. Calculate:**

- $s = 0 + 24 = 24 \, \text{m}$.

**5. Answer:**

- Displacement: $24 \, \text{m}$.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.