Constant Acceleration Models

Overview

Many H2 kinematics problems use a constant acceleration model. When this model is valid, motion can be solved efficiently using the SUVAT equations.

However, students must first check whether the assumptions are satisfied.

This page covers:

• meaning of constant acceleration
• SUVAT equations
• when SUVAT can be used
• when SUVAT cannot be used
• sign consistency
• motion under gravity
• worked examples

Main topic: Kinematics

Why It Matters

SUVAT is only reliable when the motion model is valid, so recognising the assumptions matters as much as doing the algebra.

Definition

A constant acceleration model describes motion in which acceleration stays constant over the interval being analysed.

1. What is Constant Acceleration?

Acceleration is constant when:

• magnitude remains unchanged
• direction remains unchanged

Hence velocity changes at a steady rate.

Mathematically:

$$a=\text{constant}$$

This produces a straight-line velocity-time graph.

2. Common Physical Situations

Valid Approximations

Horizontal motion with steady driving/braking

Uniform engine thrust or braking force over short interval.

Free fall near Earth’s surface

If air resistance is negligible:

$$a=g\approx 9.81{\text{m s}}^{-2}$$

downward.

Projectile components

• horizontal: constant velocity
• vertical: constant acceleration $-g$

See Projectile and Relative Motion

3. SUVAT Variables

Key Representations

Symbol	Meaning
$s$	displacement
$u$	initial velocity
$v$	final velocity
$a$	constant acceleration
$t$	time

4. SUVAT Equations

Equation 1

$$v=u+at$$

Equation 2

$$s=ut+\frac{1}{2}a{t}^2$$

Equation 3

$$v^2=u^2+2as$$

Equation 4

$$s=\frac{u+v}{2}t$$

5. Choosing the Correct Equation

Use the equation that excludes the unwanted variable.

Equation	Missing Variable
$v=u+at$	$s$
$s=ut+\frac{1}{2}a{t}^2$	$v$
$v^2=u^2+2as$	$t$
$s=\frac{u+v}{2}t$	$a$

6. Where SUVAT Comes From

Two key ideas:

From acceleration definition

$$a=\frac{v-u}{t}\Rightarrow v=u+at$$

From area under v-t graph

For constant acceleration, the v-t graph is a trapezium.

Area gives displacement:

$$s=\frac{u+v}{2}t$$

Other equations follow algebraically.

See Kinematics Graphs and Calculus

7. Conditions for Using SUVAT

Use SUVAT only when:

• acceleration is constant
• motion is in one straight line, or one resolved component
• same object throughout
• signs are used consistently

8. When SUVAT Cannot Be Used Directly

Variable Acceleration

Examples:

• strong air resistance
• acceleration changing with time
• acceleration depending on position

Curved Motion Without Resolution

Need components or other methods.

Multi-Stage Motion

Split into separate sections.

Example:

• accelerate
• travel at constant speed
• brake

Treat each segment separately.

9. Sign Convention Reminders

Choose positive direction first.

Examples:

• upward positive
• right positive

Then apply consistently.

Vertical Motion Example

If upward is positive:

$$a=-g$$

If downward is positive:

$$a=+g$$

Both are correct if consistent.

10. Motion Under Gravity

Near Earth with negligible air resistance:

$$g=9.81{\text{m s}}^{-2}$$

At highest point of upward throw:

$$v=0$$

but

$$a=-g$$

So zero velocity does not mean zero acceleration.

11. Worked Examples

Example 1: Car from Rest

A car starts from rest and accelerates uniformly at $3.0{\text{m s}}^{-2}$ for 4.0 s.

Find final velocity and displacement.

Step 1

$$v=u+at=0+3(4)=12{\text{m s}}^{-1}$$

Step 2

$$s=ut+\frac{1}{2}a{t}^2$$ $$s=0+\frac{1}{2}(3)(4^2)=24\text{m}$$

Example 2: Ball Thrown Upward

Ball thrown upward at $20{\text{m s}}^{-1}$.

Take upward positive.

Find maximum height.

At top:

$$v=0,a=-9.81$$

Use:

$$v^2=u^2+2as$$ $$0=20^2+2(-9.81)s$$ $$s=20.4\text{m}$$

Example 3: Braking Distance

Car moving at $25{\text{m s}}^{-1}$ brakes at $-5.0{\text{m s}}^{-2}$.

Find stopping distance.

$$0=25^2+2(-5)s$$ $$s=62.5\text{m}$$

12. Common Exam Pitfalls

• using SUVAT when acceleration varies
• forgetting to define positive direction
• mixing signs halfway through solution
• using $v=0$ at top and assuming $a=0$
• solving multi-stage motion as one stage
• using total distance instead of displacement

13. Quick Summary

Use SUVAT when:

• constant acceleration
• one-dimensional motion/component
• clear sign convention

Core equations:

$$v=u+at$$ $$s=ut+\frac{1}{2}a{t}^2$$ $$v^2=u^2+2as$$ $$s=\frac{u+v}{2}t$$

Related Links

Links

• Kinematics
• Kinematic Quantities and Sign Conventions
• Kinematics Graphs and Calculus
• Projectile and Relative Motion
• Projectile Motion
• Forces
• Dynamics