Speed, Displacement, Velocity, and Acceleration (Junior Cert Science): Revision Notes

Speed, Displacement, Velocity, and Acceleration

Introduction to motion

When we describe how objects move, we need to understand several related but different concepts. This note will help you understand speed, displacement, velocity, and acceleration, and how to represent motion using graphs. These concepts are fundamental to understanding motion in physics.

Speed

Speed tells us how quickly an object moves from one place to another. It is an important measurement in everyday life, from estimating journey times to road safety.

What is speed?

Speed measures how fast an object changes its position. To calculate speed, we need to know two things:

• The distance travelled (measured in metres, m)
• The time taken (measured in seconds, s)

The formula for calculating speed is:

$\text{speed} = \frac{\text{distance (m)}}{\text{time (s)}}$

The units of speed are metres per second, written as $\text{m/s}$ or $\text{m s}^{-1}$.

Different speeds in our world

Different objects and activities have very different speeds. Here are some examples:

Activity/Object	Speed ($\text{m s}^{-1}$)
Walking	$1.5$
Car at $100$ km/hour	$28$
Cheetah	$33$
Falcon diving	$90$
Sound	$340$
Jet fighter	$890$
Space shuttle	$8,333$
Light travelling through a vacuum	$300,000,000$

Worked example: calculating speed

Displacement

While distance tells us how far an object has travelled, displacement gives us more specific information about the object's change in position.

What is displacement?

Displacement is the distance travelled in a given direction.

This is an important distinction. An object might travel a long distance but end up close to where it started. In such cases, the displacement would be small even though the distance travelled was large.

Displacement depends only on start and finish positions

This world map shows an excellent example. A boat travelling from London to Mumbai could take three different routes:

• Route A through the Suez Canal: $7,000$ km
• Route B alternative route via Suez Canal: $11,600$ km
• Route C around the Cape of Good Hope: $19,800$ km

The distance travelled is very different for each route. However, the displacement at the end of each journey is approximately $7,000$ km (the straight-line distance from London to Mumbai in a given direction). This shows that displacement is independent of the pathway taken.

Velocity

Velocity is closely related to speed, but includes information about direction.

What is velocity?

Velocity is speed in a given direction. This means that to describe velocity fully, you must state both:

• How fast the object is moving
• The direction in which it is moving

The formula for velocity is:

$\text{velocity} = \frac{\text{change in displacement}}{\text{time taken}}$

The difference between speed and velocity

The terms speed and velocity are often used interchangeably in everyday conversation, but in physics they have different meanings:

• Speed only tells you how fast something is moving
• Velocity tells you how fast something is moving AND in which direction

Worked example: calculating velocity

Distance-time graphs

Graphs provide a powerful visual way to represent and analyse motion. Distance-time graphs show how the distance travelled by an object changes over time.

How to draw a distance-time graph correctly

When drawing any graph in science, follow these four important steps:

1. Label each axis with the quantity and its units
2. Draw a correct scale on each axis (make sure the scale is even and clearly marked)
3. Plot each point accurately using the data provided
4. Join the points appropriately (with a straight line if the points follow a linear pattern, or a smooth curve if they follow a curved pattern)

Example data for a walker

Let's look at data collected for a person walking at constant speed:

Time (s)	$0$	$1$	$2$	$3$	$4$	$5$
Distance (m)	$0$	$2$	$4$	$6$	$8$	$10$

This data shows that the walker travels $2$ metres every second, indicating constant speed.

Reading values from a distance-time graph

To find values from a distance-time graph:

1. Find the value you want on one axis
2. Draw a horizontal line across to meet the plotted line
3. Draw a vertical line down to meet the time axis
4. Read the corresponding value

This technique allows you to find the time for any given distance, or the distance at any given time.

Calculating speed from a distance-time graph

One of the most useful features of distance-time graphs is that the slope (gradient) of the line tells you the speed.

To calculate speed from the graph:

1. Choose two points on the line
2. Find the vertical change (change in distance)
3. Find the horizontal change (change in time)
4. Calculate: $\text{speed} = \frac{\text{change in distance}}{\text{change in time}}$

Interpreting different slopes on distance-time graphs

Real journeys often involve changing speeds or periods of rest. Here's what different features of distance-time graphs tell us:

• Steep slope = high speed (covering more distance in less time)
• Gentle slope = low speed (covering less distance in the same time)
• Horizontal line (no slope) = zero speed (the object has stopped)
• Downward slope = negative speed (moving back towards the starting point)
• Changing slope = changing speed (accelerating or decelerating)

Worked example: interpreting a distance-time graph

Consider a student carrying out a fitness test, walking and running at different speeds. The distance travelled over time was recorded:

Time (s)	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
Distance (m)	$0$	$4$	$8$	$12$	$14$	$16$	$16$	$16$	$12$	$8$	$4$

Using motion sensors in experiments

Modern technology allows us to measure motion very accurately. A motion sensor can be connected to a computer to plot graphs of distance versus time automatically.

Acceleration

When an object's velocity changes, we say it is accelerating. Acceleration is one of the most important concepts in physics.

What is acceleration?

Acceleration is the change in velocity in one second.

More formally, acceleration is the rate of change of velocity. This means it measures how quickly velocity changes over time.

The formula for acceleration is:

$\text{acceleration} = \frac{\text{change in velocity}}{\text{time taken}}$

The units of acceleration are metres per second per second, written as $\text{m/s}^2$ or $\text{m s}^{-2}$.

Understanding the units of acceleration

Positive and negative acceleration

• Positive acceleration means the velocity is increasing (speeding up)
• Negative acceleration (also called deceleration) means the velocity is decreasing (slowing down)

Worked example: calculating acceleration

Velocity-time graphs

Just as distance-time graphs help us visualise and analyse motion, velocity-time graphs show how velocity changes over time.

Imagine looking at a car's speedometer every second over $8$ seconds. As the car accelerates, the needle moves to show increasing velocity. A velocity-time graph captures this information in a visual format.

Example data for an accelerating car

Time (s)	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
Velocity ($\text{m s}^{-1}$)	$0$	$5$	$10$	$15$	$20$	$25$	$25$	$25$	$25$

This data shows two distinct phases:

• 0 to 4 seconds: Velocity is increasing (positive acceleration)
• 4 to 8 seconds: Velocity remains constant at $25 \text{ m s}^{-1}$ (zero acceleration)

Interpreting velocity-time graphs

Key features to understand:

• Upward slope = positive acceleration (velocity increasing)
• Steeper slope = greater acceleration
• Horizontal line = zero acceleration (constant velocity)
• Downward slope = negative acceleration/deceleration (velocity decreasing)

Calculating acceleration from a velocity-time graph

To find acceleration from a velocity-time graph, calculate the slope:

$\text{acceleration} = \text{slope} = \frac{\text{change in velocity}}{\text{change in time}}$

Measuring acceleration experimentally

Results: When the experiment is repeated with different incline angles:

• Steeper incline: slope = $0.54 \text{ m s}^{-2}$ (greater acceleration)
• Gentler incline: slope = $0.39 \text{ m s}^{-2}$ (less acceleration)

This demonstrates that the steeper the incline, the greater the acceleration.

Sample data from trolley experiment

Time (s)	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$
Velocity ($\text{m s}^{-1}$)	$0$	$2$	$4$	$6$	$8$	$10$	$12$	$12$

The velocity increases by $2 \text{ m s}^{-1}$ every second for the first $6$ seconds, then remains constant. This gives an acceleration of $2 \text{ m s}^{-2}$ during the first phase.

Acceleration due to gravity

One of the most important accelerations in physics is the acceleration due to gravity, which affects all falling objects.

Free fall

When an object is dropped from a height, it accelerates towards the Earth due to the force of gravity. The velocity increases by approximately $10 \text{ m s}^{-1}$ every second.

Here's what happens when a ball is dropped from a cliff:

Time (s)	$0$	$1$	$2$	$3$	$4$	$5$
Velocity ($\text{m s}^{-1}$)	$0$	$10$	$20$	$30$	$40$	$50$

The effect of air resistance

In reality, air resistance affects how objects fall:

• In air: A feather falls slowly because air resistance opposes its motion. The large surface area of the feather experiences significant air resistance
• In a vacuum: With no air resistance, both a feather and a coin fall at exactly the same rate, accelerating at $10 \text{ m s}^{-2}$

Important points about falling objects

• In the first tube (with air), the feather falls at an angle and more slowly due to air resistance
• In the second tube (vacuum), the feather falls straight down at the same rate as any other object
• This proves that gravity causes the same acceleration for all objects
• A stone will fall faster than a feather in air, but they accelerate at the same rate in a vacuum

Practice with graphs

Let's look at a complete journey represented on a distance-time graph. This graph shows distance (in metres) against time (in seconds) with five marked points A, B, C, D, and E.

Analysing the journey:

• Point A to B: The person travels away from the starting point. The slope tells us the speed
• Point B to C: The slope changes, indicating a different speed
• Point C to D: The steepest section - this is where the person is travelling fastest
• Point D to E: A downward slope showing the person returning towards the starting point

By calculating the slope of each section, you can determine the speed during each phase of the journey.

Remember!