Physics | Position and movement (kinematics) 

Position and Movement (Kinematics) - Your Study Guide!

Hey there! Welcome to the exciting world of Kinematics. That's just a fancy word for describing how things move. Ever wondered how fast a sprinter runs, how a thrown ball travels through the air, or how to read a car's speedometer? That's all kinematics! In this chapter, we'll learn the language of motion. It’s the foundation for almost everything else in Physics, so getting a good grip on these ideas will make your journey much smoother. Don't worry if some concepts seem tricky at first – we'll break everything down with simple examples you see every day. Let's get moving!

1. The Starting Point: Position, Distance, and Displacement

Before we can describe motion, we need to know where something is and where it's going. This sounds simple, but in Physics, we need to be very precise!

Position (s)

Position is simply an object's location relative to a reference point (we often call this the 'origin'). It's like giving an address. For example, "My friend is 5 metres to the right of the school gate." Here, the school gate is the origin.

Distance vs. Displacement

This is one of the most important distinctions in kinematics. It’s a common place to get confused, so let's make it super clear!

• Distance (d): This is the total path length travelled. Think of it as the reading on a car's odometer or your fitness tracker. It tells you "how much ground you covered." Distance only has a size (magnitude), so it doesn't care about direction.

• Displacement (s): This is the object's change in position. It's the straight-line distance from the starting point to the ending point, and it includes the direction. It tells you "how far you are from where you started."

Analogy: Your Walk to the Shops

Imagine you walk from your home 400 m East to a shop, and then you walk back 100 m West to a postbox.

- Distance travelled: You walked 400 m + 100 m = 500 m. This is the total path you took.

- Displacement: Your starting point was home. Your end point is the postbox, which is 400 m - 100 m = 300 m East of your home. Displacement must have a direction!

Scalars and Vectors

The difference between distance and displacement introduces us to two types of physical quantities:

• Scalars are quantities that have magnitude (size) only.
Examples: distance (500 m), speed (20 m/s), time (15 s), mass (5 kg).

• Vectors are quantities that have both magnitude and direction.
Examples: displacement (300 m East), velocity (20 m/s North), acceleration (9.81 m/s² downwards).

Memory Aid!

Think "Scalar" for "Size only".
Think "Vector" for "Velocity" (which you'll soon learn has direction!).

Key Takeaway

Distance is the total path length (a scalar). Displacement is the straight-line change in position, including direction (a vector).

2. How Fast?: Speed and Velocity

Now that we know about distance and displacement, we can describe how quickly an object covers that ground.

Average Speed

Average speed is a scalar quantity that tells you the total distance travelled divided by the total time taken. It doesn't care about speeding up, slowing down, or changing direction.

$$ \text{Average speed} = \frac{\text{Total distance travelled}}{\text{Total time taken}} $$

Average Velocity

Average velocity is a vector quantity. It's the total displacement divided by the total time taken. Since displacement has a direction, velocity does too!

$$ \text{Average velocity} = \frac{\text{Total displacement}}{\text{Total time taken}} = \frac{\Delta s}{\Delta t} $$

(The symbol Δ means 'change in').

Common Mistake Alert!

Never use distance to calculate velocity, or displacement to calculate speed. They are different concepts! In our 'walk to the shops' example, if the walk took 100 seconds:
- Average speed = 500 m / 100 s = 5 m/s
- Average velocity = 300 m East / 100 s = 3 m/s East
See how different they are!

Instantaneous Speed and Velocity

What about the speed at a single moment in time?
• Instantaneous speed is the speed at a particular instant. A car's speedometer shows your instantaneous speed.
• Instantaneous velocity is the velocity (speed and direction) at a particular instant.

Key Takeaway

Speed is the rate of change of distance (scalar). Velocity is the rate of change of displacement (vector). An object can have a constant speed but a changing velocity if it changes direction (e.g., a car turning a corner).

3. Visualising Motion: Graphs of Motion

Graphs are a powerful way for physicists to "see" motion. You need to be an expert at reading and interpreting three types of graphs.

Displacement-time graphs (s-t graphs)

These graphs show an object's position (displacement) at different times.

• The gradient (slope) of an s-t graph gives the velocity.

- A horizontal line means the displacement isn't changing. The object is at rest (velocity = 0).
- A straight line with a positive slope means the object is moving with a constant positive velocity.
- A straight line with a negative slope means the object is moving with a constant negative velocity (moving back towards the origin).
- A curved line means the slope is changing, so the velocity is changing (the object is accelerating).

Velocity-time graphs (v-t graphs)

These graphs show an object's velocity at different times. They are super useful!

• The gradient (slope) of a v-t graph gives the acceleration.
• The area under a v-t graph gives the displacement.

- A horizontal line means the velocity is constant. The object is in uniform motion (acceleration = 0).
- A straight line with a positive slope means the object is moving with constant positive acceleration.
- A straight line with a negative slope means the object is moving with constant negative acceleration (decelerating).
- The area of the shape(s) between the line and the time-axis gives the total displacement. Remember that area below the axis represents negative displacement!

Acceleration-time graphs (a-t graphs)

These graphs show an object's acceleration at different times.

• The area under an a-t graph gives the change in velocity (Δv).

In HKDSE Physics, you'll mostly deal with constant acceleration.
- A horizontal line at a > 0 means constant positive acceleration.
- A horizontal line at a = 0 means constant velocity.
- A horizontal line at a < 0 means constant negative acceleration.

Quick Review: What Graphs Tell You

Graph Type
Gradient (Slope) tells you...
Area under graph tells you...

s-t graph
Velocity (v)
(Nothing useful for us)

v-t graph
Acceleration (a)
Displacement (s)

a-t graph
(Nothing useful for us)
Change in velocity (Δv)

4. Changing Gears: Acceleration

Most things in the real world don't move at a constant velocity. They speed up and slow down. This change in velocity is called acceleration.

Acceleration (a) is defined as the rate of change of velocity. Since velocity is a vector, acceleration is also a vector.

$$ a = \frac{\text{change in velocity}}{\text{time taken}} = \frac{v - u}{t} $$

Where:
• a = acceleration (in m s⁻²)
• v = final velocity (in m s⁻¹)
• u = initial velocity (in m s⁻¹)
• t = time taken (in s)

An object is accelerating if its speed changes OR its direction of motion changes. A car turning a corner at a constant speed is still accelerating because its direction (and therefore velocity) is changing!

Deceleration

Deceleration means slowing down. It's just a word for negative acceleration in the direction of motion. For example, if a car moving in the positive direction is slowing down, its acceleration is negative.

Did you know?

The acceleration you feel in a sports car is often measured in 'g's. One 'g' is about 9.81 m s⁻², the acceleration of a falling object near Earth. Some drag racers experience over 4g's!

Key Takeaway

Acceleration is any change in velocity (speed or direction) over time. A positive acceleration means velocity is becoming more positive. A negative acceleration means velocity is becoming more negative.

5. The Super Formulas: Equations of Uniformly Accelerated Motion

When an object moves with constant acceleration, we can use a set of four amazing equations to solve problems. These are often called the 'suvat' equations.

IMPORTANT: Only use these for CONSTANT (uniform) acceleration!

Here are the variables:

s = displacement
u = initial velocity
v = final velocity
a = acceleration
t = time

The Four Equations:

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

How to Solve Problems Step-by-Step:

1. List your variables: Write down s, u, v, a, t.
2. Read the question: Fill in the values you know. Identify the value you need to find.
3. Choose the right equation: Find the equation that contains your known variables and the one unknown you want to find. (Each equation leaves out one of the five variables).
4. Substitute and Solve: Plug in the numbers and calculate the answer. Don't forget the units!

Example Walkthrough

A car starts from rest and accelerates uniformly at 2 m s⁻² for 5 s. Find its final velocity and the distance it travelled.

1 & 2. List variables:
s = ?
u = 0 m s⁻¹ ('from rest' means initial velocity is zero)
v = ?
a = 2 m s⁻²
t = 5 s

3. Find final velocity (v): We know u, a, t and want v. The best equation is $$v = u + at$$.
4. Solve for v: $$v = 0 + (2)(5) = 10 \text{ m s⁻¹}$$.

3. Find distance (s): We know u, a, t and want s. Let's use $$s = ut + \frac{1}{2}at^2$$.
4. Solve for s: $$s = (0)(5) + \frac{1}{2}(2)(5)^2 = 0 + \frac{1}{2}(2)(25) = 25 \text{ m}$$.

6. Down to Earth: Vertical Motion Under Gravity

One of the most common examples of constant acceleration is an object falling freely near the Earth's surface. This is called free fall.

Acceleration Due to Gravity (g)

In the absence of air resistance, all objects, regardless of their mass, fall with the same constant acceleration. This is the acceleration due to gravity, represented by the symbol g.

• On Earth, g ≈ 9.81 m s⁻² (sometimes approximated as 10 m s⁻² in problems).
• This acceleration is always directed downwards, towards the centre of the Earth.

Using the Equations of Motion for Free Fall

We can use the same four equations of motion! We just need to be careful with signs.

Sign Convention: It's easiest to be consistent. A common choice is:
• Upwards is the positive (+) direction.
• Downwards is the negative (-) direction.

This means:
- The acceleration 'a' will always be a = -g = -9.81 m s⁻².
- A ball thrown upwards has a positive initial velocity (+u).
- A ball falling downwards has a negative velocity (-v).

Common Mistake Alert!

When you throw a ball straight up, at its highest point, its instantaneous velocity is zero (v = 0), but its acceleration is NOT zero! It is still -9.81 m s⁻² because gravity is still pulling it down. This is what makes it start moving down again.

The Effect of Air Resistance

In the real world, we can't ignore air resistance (or drag).
• Air resistance is a force that opposes motion through the air.
• It increases as an object's speed increases.
• When a falling object's speed increases, its air resistance increases until it becomes equal in size to the object's weight. At this point, the net force is zero, and the object stops accelerating. It falls at a constant maximum velocity called terminal velocity.

Think of a skydiver: they accelerate at first, but then reach a terminal velocity of about 200 km/h. When they open their parachute, the increased air resistance slows them down to a new, much lower terminal velocity, allowing them to land safely.

Key Takeaway

Free fall motion is just a special case of uniformly accelerated motion where a = -g (if up is positive). Remember to be consistent with your signs! Air resistance limits the speed of falling objects, leading to a terminal velocity.