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