📏 Position, Displacement, and Distance

Part 1 of 7 — One-Dimensional Motion

Kinematics is the study of how things move — without worrying about why they move. Before we can describe motion mathematically, we need precise definitions of where an object is and how far it has traveled.

In this lesson, we'll distinguish between three foundational concepts:

• Position — where an object is
• Distance — how far it has traveled (total path length)
• Displacement — how far and in what direction it has moved from its starting point

Position and Coordinate Systems

Position ($x$) describes an object's location along a number line relative to a chosen origin (the zero point).

Key Ideas

• Position is measured in meters (m) in SI units
• You must choose a coordinate system: a reference point (origin) and a positive direction
• Position can be positive or negative depending on which side of the origin the object is on

Example

If we set the origin at a mailbox on a straight road:

• A car 50 m to the right: $x = +50$ m
• A car 30 m to the left: $x = -30$ m

Important: The choice of origin is arbitrary — different observers can choose different origins, but the physics doesn't change.

Displacement

Displacement ($\Delta x$) is the change in position:

$\Delta x = x_f - x_i$

where $x_f$ is the final position and $x_i$ is the initial position.

Properties of Displacement

Example

A runner starts at $x_i = 2$ m and finishes at $x_f = 8$ m:

$\Delta x = 8 - 2 = +6 \text{ m}$

The positive sign tells us the runner moved in the positive direction.

Distance vs. Displacement

This is one of the most important distinctions in kinematics!

Example

A student walks 4 m east, then 3 m west:

• Distance = $4 + 3 = 7$ m (total path traveled)
• Displacement = $4 - 3 = +1$ m east (net change in position)

When Are They Equal?

Distance equals the magnitude of displacement only when the object moves in a straight line without changing direction.

Concept Check — Position, Distance, and Displacement 🎯

Displacement Calculations 🧮

1. A car drives from position $x_i = 10$ m to $x_f = -15$ m. What is the displacement? (include sign, in meters)

2. A hiker walks 6 km north, then 2 km south. What is the total distance traveled? (in km)

3. For the same hiker in problem 2, what is the magnitude of the displacement? (in km)

Classify Each Quantity 🔍

Exit Quiz — Position, Distance & Displacement ✅

🏃 Average Velocity and Average Speed

Part 2 of 7 — One-Dimensional Motion

Now that we understand position and displacement, we can describe how fast an object moves. There are two closely related but distinct quantities: average velocity and average speed.

Average Velocity

Average velocity is the rate of change of position — displacement divided by elapsed time:

$v_{\text{avg}} = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}$

Properties

Example

A car travels from $x_i = 20$ m at $t_i = 0$ s to $x_f = 80$ m at $t_f = 4$ s:

$v_{\text{avg}} = \frac{80 - 20}{4 - 0} = \frac{60}{4} = 15 \text{ m/s}$

Average Speed

Average speed is the total distance traveled divided by the elapsed time:

$\text{average speed} = \frac{\text{total distance}}{\Delta t}$

Velocity vs. Speed

Example

A runner goes 100 m east in 10 s, then 60 m west in 6 s:

• $v_{\text{avg}} = \frac{100 - 60}{10 + 6} = \frac{40}{16} = 2.5$ m/s (east)
• Average speed $= \frac{100 + 60}{16} = \frac{160}{16} = 10$ m/s

Key insight: Average speed ≥ |average velocity|, with equality only when the object doesn't reverse direction.

Concept Check — Velocity and Speed 🎯

Velocity and Speed Calculations 🧮

1. A bus travels 240 m east in 30 s. What is the average velocity? (in m/s)

2. A delivery truck drives 80 km in 2 hours, then returns 80 km in 3 hours. What is the average speed for the entire trip? (in km/h, round to 3 significant figures)

3. For the same delivery truck, what is the magnitude of the average velocity for the entire trip? (in km/h)

Quick Concept Review 🔍

Exit Quiz — Average Velocity & Speed ✅

⚡ Acceleration

Part 3 of 7 — One-Dimensional Motion

So far we know how to describe where an object is (position) and how fast it moves (velocity). Now we need to describe how the velocity itself changes — that's acceleration.

Acceleration is the rate of change of velocity, just as velocity is the rate of change of position.

Defining Acceleration

Average acceleration is the change in velocity divided by the elapsed time:

$a_{\text{avg}} = \frac{\Delta v}{\Delta t} = \frac{v_f - v_i}{t_f - t_i}$

Properties

Example

A car goes from $v_i = 10$ m/s to $v_f = 30$ m/s in 4 s:

$a_{\text{avg}} = \frac{30 - 10}{4} = \frac{20}{4} = 5 \text{ m/s}^2$

This means the car's velocity increases by 5 m/s every second.

The Sign of Acceleration

A common misconception: negative acceleration does NOT always mean slowing down!

What matters is the relationship between the signs of velocity and acceleration:

Rule of Thumb

• Same sign (velocity and acceleration): object speeds up
• Opposite signs: object slows down

Example

A car moving west (negative direction) at $v_i = -20$ m/s accelerates to $v_f = -30$ m/s in 5 s:

$a = \frac{-30 - (-20)}{5} = \frac{-10}{5} = -2 \text{ m/s}^2$

Both velocity and acceleration are negative → the car is speeding up (going faster in the negative direction).

Concept Check — Acceleration 🎯

Acceleration Calculations 🧮

1. A train accelerates from rest to 36 m/s in 12 s. What is the average acceleration? (in m/s²)

2. A ball rolling at 14 m/s comes to rest in 7 s. What is the average acceleration? (in m/s², include the sign)

3. An object has acceleration $a = 4$ m/s² and initial velocity $v_i = 3$ m/s. What is the velocity after 5 s? (in m/s)

Acceleration Concepts 🔍

Exit Quiz — Acceleration ✅

📐 The Kinematic Equations

Part 4 of 7 — One-Dimensional Motion

When acceleration is constant, we can derive a powerful set of equations that relate position, velocity, acceleration, and time. These are the kinematic equations — the core tools for solving 1D motion problems.

The Big Three Kinematic Equations

For constant acceleration along a straight line:

Equation 1: Velocity-Time

$v = v_0 + at$

Equation 2: Position-Time

$x = x_0 + v_0 t + \frac{1}{2}at^2$

Equation 3: Velocity-Position (no time)

$v^2 = v_0^2 + 2a(x - x_0)$

Or equivalently: $v^2 = v_0^2 + 2a\Delta x$

Variable Summary

Important: These equations work ONLY for constant acceleration. If $a$ changes, you need calculus or break the problem into intervals of constant $a$.

Choosing the Right Equation

Each equation is missing one variable. Choose based on what you know and what you need:

Problem-Solving Strategy

1. List knowns — identify which variables you have
2. Identify the unknown — what are you solving for?
3. Choose the equation — pick the one that contains your unknown and your knowns
4. Solve algebraically — rearrange and substitute
5. Check units and sign — does your answer make physical sense?

Worked Example

Problem: A car starts from rest and accelerates at $3$ m/s² for $8$ s. Find (a) the final velocity and (b) the distance traveled.

Solution:

Known: $v_0 = 0$, $a = 3$ m/s², $t = 8$ s, $x_0 = 0$

(a) Use $v = v_0 + at$: $v = 0 + (3)(8) = 24 \text{ m/s}$

(b) Use $x = x_0 + v_0t + \frac{1}{2}at^2$: $x = 0 + 0(8) + \frac{1}{2}(3)(8)^2 = \frac{1}{2}(3)(64) = 96 \text{ m}$

Check: We can verify with $v^2 = v_0^2 + 2a\Delta x$: $24^2 = 0 + 2(3)(96) = 576$ ✓

Concept Check — Kinematic Equations 🎯

Kinematic Equation Practice 🧮

1. A motorcycle accelerates from $v_0 = 5$ m/s at $a = 2$ m/s² for $t = 6$ s. What is the final velocity? (in m/s)

2. A plane needs to reach 80 m/s to take off. If it starts from rest and accelerates at 4 m/s², what runway length does it need? (in meters)

3. A car traveling at 30 m/s brakes with $a = -6$ m/s². How long does it take to stop? (in seconds)

Equation Selection Practice 🔍

Exit Quiz — Kinematic Equations ✅

🍎 Free Fall

Part 5 of 7 — One-Dimensional Motion

Free fall is a special case of constant acceleration where the only force acting on an object is gravity. Near Earth's surface, all objects in free fall experience the same acceleration — regardless of mass!

$g = 9.8 \text{ m/s}^2 \approx 10 \text{ m/s}^2$

This is one of the most elegant results in physics, first demonstrated by Galileo.

Setting Up Free Fall Problems

Sign Convention (standard)

Taking upward as positive:

• $a = -g = -9.8$ m/s²
• Upward velocities are positive
• Downward velocities are negative

The Kinematic Equations for Free Fall

Key Facts

• At the highest point, $v = 0$ (momentarily at rest)
• The acceleration is always $-g$, even at the top
• Objects spend equal time going up and coming down (if launched and caught at same height)

Dropped Objects

When an object is dropped from rest: $v_0 = 0$

The equations simplify to:

• $v = -gt$
• $y = y_0 - \frac{1}{2}gt^2$
• $v^2 = -2g\Delta y$

Example: Dropping a stone

A stone is dropped from a 45 m cliff. How long does it take to hit the ground?

Taking $y_0 = 45$ m and the ground as $y = 0$:

$0 = 45 - \frac{1}{2}(9.8)t^2$

$t^2 = \frac{2(45)}{9.8} = 9.18$

$t = 3.03 \text{ s}$

What is its speed at impact?

$v = -gt = -(9.8)(3.03) = -29.7 \text{ m/s}$

Speed = $|v| = 29.7$ m/s (about 107 km/h!)

Objects Thrown Upward

When thrown upward with $v_0 > 0$:

Finding Maximum Height

At the top, $v = 0$:

$0 = v_0^2 - 2g\Delta y$

$\Delta y_{\text{max}} = \frac{v_0^2}{2g}$

Finding Total Time (launched and caught at same height)

Time up = time down, and at the top $v = 0$:

$0 = v_0 - gt_{\text{up}}$

$t_{\text{up}} = \frac{v_0}{g}$

$t_{\text{total}} = 2t_{\text{up}} = \frac{2v_0}{g}$

Example

A ball is thrown upward at 19.6 m/s:

• Max height: $\Delta y = 19.6^2 / (2 \times 9.8) = 19.6$ m
• Time up: $t = 19.6/9.8 = 2.0$ s
• Total time: $4.0$ s

Concept Check — Free Fall 🎯

Free Fall Calculations 🧮

Use $g = 9.8$ m/s².

1. A ball is dropped from a height of 19.6 m. How long does it take to hit the ground? (in seconds)

2. An object is thrown straight up at 29.4 m/s. What maximum height does it reach? (in meters)

3. A stone is dropped from rest. What is its speed after 3 seconds of falling? (in m/s)

Round all answers to 3 significant figures.

Free Fall Concepts 🔍

Exit Quiz — Free Fall ✅

🛠️ Problem-Solving Workshop

Part 6 of 7 — One-Dimensional Motion

This lesson is all about practice. We'll work through a variety of 1D kinematics problems, building your confidence with the kinematic equations, free fall, and multi-step problems.

Problem-Solving Framework

1. Draw a diagram — sketch the situation, label known values
2. Define your coordinate system — choose an origin and positive direction
3. List knowns and unknowns — organize with a table
4. Select the right equation — match to your knowns/unknowns
5. Solve and check — does the answer make physical sense?

Warm-Up Problems 🎯

Multi-Step Problems 🧮

1. A car accelerates from 10 m/s to 30 m/s over 200 m. What is the acceleration? (in m/s²)

2. A ball is thrown downward from a 50 m building at 5 m/s. How long until it hits the ground? (in seconds, round to 3 significant figures; use $g = 10$ m/s²)

3. A police car starts from rest and accelerates at 3 m/s². A speeder passes at a constant 24 m/s at the same moment. How long until the police car catches the speeder? (in seconds)

Free Fall Applications 🎯

Challenge Problems 🏆

1. Two cars start from rest at the same point. Car A accelerates at 2 m/s² and Car B accelerates at 3 m/s². After 10 s, how much farther has Car B traveled than Car A? (in meters)

2. A ball is dropped from 80 m. At the same instant, a ball is thrown upward from the ground at 20 m/s. At what height do they meet? (in meters, use $g = 10$ m/s²; round to 3 significant figures)

Exit Quiz — Problem Solving ✅

🎓 Synthesis & AP Review

Part 7 of 7 — One-Dimensional Motion

Let's bring together everything from this unit: position, displacement, velocity, acceleration, kinematic equations, and free fall. This is your comprehensive review to prepare for AP-level questions.

Complete Summary

Core Definitions

Kinematic Equations (constant $a$)

1. $v = v_0 + at$
2. $x = x_0 + v_0t + \frac{1}{2}at^2$
3. $v^2 = v_0^2 + 2a\Delta x$

Free Fall

• $a = -g = -9.8$ m/s² (taking up as positive)
• All objects fall at the same rate (ignoring air resistance)
• At maximum height: $v = 0$

Common AP Traps

• Negative acceleration ≠ slowing down (depends on velocity sign)
• Distance ≠ displacement (especially for round trips)
• Acceleration at the top of a throw = $-g$, NOT zero

AP-Style Questions — Set 1 🎯

AP-Style Calculations 🧮

1. A car accelerates uniformly from 10 m/s to 25 m/s over a distance of 175 m. How long does this take? (in seconds)

2. A ball is thrown upward at 24.5 m/s. How high above the launch point is it after 2 s? (in meters; use $g = 9.8$ m/s²)

3. A train brakes from 36 m/s to 16 m/s over 8 s. What distance does it cover while braking? (in meters)

Round all answers to 3 significant figures.

Conceptual Mastery Check 🔍

Final AP Review ✅