Physics (9630) | Motion along a straight line

Physics (9630): Study Notes – Motion along a straight line (3.2.3)

Welcome to the exciting world of kinematics! This chapter, Motion along a straight line (sometimes called one-dimensional motion), is the absolute foundation of mechanics. Everything we learn about forces, energy, and momentum later depends on understanding how things move, where they are, and how fast they change speed.

Don't worry if the formulas look complicated at first. We will break down every concept step-by-step. Let's get moving!

1. Describing Position and Movement: Scalars and Vectors

Before we define motion, we must distinguish between two fundamental types of physical quantities:

1.1 Key Definitions: Distance vs. Displacement

In Physics, the words we use for motion have very specific meanings. It all starts with whether a quantity is a scalar or a vector.

Scalars are quantities defined only by their magnitude (size).

• Examples: Distance, Speed, Mass, Time, Energy.

Vectors are quantities defined by both their magnitude and their direction.

• Examples: Displacement, Velocity, Acceleration, Force, Momentum.

Displacement (\(s\)) vs. Distance:

Imagine you walk 5 km North, then 3 km South.

• Distance: The total length of the path taken (5 km + 3 km = 8 km). (A scalar)
• Displacement (\(s\)): The shortest path from the start point to the end point, including direction (5 km N - 3 km S = 2 km North). (A vector)

Memory Aid: Displacement is the 'direct' route from A to B.

1.2 Key Definitions: Speed vs. Velocity

This follows the same logic as distance and displacement.

Speed is the rate of change of distance (a scalar).

Velocity (\(v\)) is the rate of change of displacement (a vector).

The definitions allow us to calculate average velocity:

\[ v_{\text{avg}} = \frac{\text{Change in displacement}}{\text{Change in time}} = \frac{\Delta s}{\Delta t} \]

Instantaneous Velocity is the velocity measured at a single, specific point in time (like the speedometer reading on your car right now). In contrast, average velocity is measured over a long time interval.

Key Takeaway: Always check if the quantity requires direction. If it does, it's a vector (Displacement, Velocity, Acceleration).

2. Acceleration

Acceleration describes how quickly your velocity changes. Crucially, because velocity is a vector, acceleration can mean a change in speed, or a change in direction (though in straight-line motion, we focus on speed change).

2.1 Definition and Formula

Acceleration (\(a\)) is the rate of change of velocity.

Its units are metres per second squared (\( \text{m\,s}^{-2} \)).

\[ a = \frac{\text{Change in velocity}}{\text{Time taken}} = \frac{\Delta v}{\Delta t} \]

where \(\Delta v = v - u\):

• \(v\) is the final velocity
• \(u\) is the initial velocity

Did you know? If an object is slowing down, its acceleration is opposite to its direction of motion. This is often called deceleration or negative acceleration. If a car is moving East and slowing down, its acceleration vector points West.

2.2 Uniform and Non-Uniform Acceleration

• Uniform Acceleration: The velocity changes by the same amount every second. This is the simplest type of motion, and all the "SUVAT" equations (Section 4) apply only to this case.
• Non-Uniform Acceleration: The rate of velocity change is constantly fluctuating (e.g., pressing and releasing the accelerator pedal in a car).

3. Graphical Representation of Motion

Graphs are essential tools in physics for visualizing motion and calculating key quantities. You must know what the gradient and the area under the graph represent for each type of motion graph.

3.1 Displacement-Time (\(s\)-t) Graphs

These graphs plot the position (displacement) of an object over time.

• Gradient: The gradient of an \(s\)-t graph gives the velocity (\(v = \frac{\Delta s}{\Delta t}\)).
• A straight line means constant velocity (zero acceleration).
• A curved line means changing velocity (acceleration or deceleration).

3.2 Velocity-Time (\(v\)-t) Graphs (The Most Important!)

These graphs plot the velocity of an object over time.

• Gradient: The gradient of a \(v\)-t graph gives the acceleration (\(a = \frac{\Delta v}{\Delta t}\)).
  • A straight, non-zero gradient means uniform acceleration.
  • A zero gradient means constant velocity (zero acceleration).
• Area under the graph: The area under a \(v\)-t graph gives the displacement (\(s\)).

Analogy: Think of a road trip. If your velocity is \(20 \, \text{m\,s}^{-1}\) for \(10\) seconds, you covered \(20 \times 10 = 200\) metres. This is exactly calculating the area of a rectangle on the \(v\)-t graph.

3.3 Acceleration-Time (\(a\)-t) Graphs

These graphs plot the acceleration of an object over time.

• Area under the graph: The area under an \(a\)-t graph gives the change in velocity (\(\Delta v\)).

Key Takeaway: For the \(v\)-t graph, remember: Gradient = Acceleration. Area = Displacement. (G.A. A.D.)

4. The Equations of Uniform Acceleration (SUVAT)

When acceleration is constant (uniform), we can use four simple equations to solve any straight-line motion problem. These are often called the SUVAT equations.

4.1 The SUVAT Variables

• s = Displacement (\(\text{m}\))
• u = Initial velocity (\(\text{m\,s}^{-1}\))
• v = Final velocity (\(\text{m\,s}^{-1}\))
• a = Uniform acceleration (\(\text{m\,s}^{-2}\))
• t = Time taken (\(\text{s}\))

4.2 The Four Equations

You must be able to recall and use these equations:

Equation 1 (No \(s\)):
\[ v = u + at \]

Equation 2 (No \(a\)):
\[ s = \left(\frac{u+v}{2}\right)t \]

Equation 3 (No \(v\)):
\[ s = ut + \frac{1}{2}at^2 \]

Equation 4 (No \(t\)):
\[ v^2 = u^2 + 2as \]

4.3 Strategy for Solving SUVAT Problems (Step-by-Step)

When solving a problem involving uniform acceleration:

1. List: Write down the five SUVAT variables (\(s, u, v, a, t\)).
2. Identify: Fill in the values you are given, including the unit. Remember that "starts from rest" means \(u=0\), and "comes to a stop" means \(v=0\).
3. Target: Identify the variable you need to find (e.g., find \(t\)).
4. Select: Choose the SUVAT equation that uses your three known variables and your one target variable (i.e., the equation that omits the unknown fifth variable).
5. Calculate: Substitute the values and solve.

Common Mistake to Avoid: Never mix units! Ensure all quantities are in standard SI units (metres, seconds, \(\text{m\,s}^{-1}\), \(\text{m\,s}^{-2}\)) before substituting into the equations.

5. Motion under Gravity (Free Fall)

The motion of an object thrown or dropped near the Earth's surface is a perfect example of uniform acceleration, provided we ignore air resistance.

5.1 Acceleration due to Gravity (\(g\))

The Earth exerts a gravitational pull, causing all objects to accelerate downwards at a constant rate, denoted by \(g\).

• The standard value of \(g\) is approximately \(9.81 \, \text{m\,s}^{-2}\) (often simplified to \(10 \, \text{m\,s}^{-2}\) in some contexts, but use the value given in your data booklet or question).
• This value is always directed downwards.

5.2 Applying SUVAT to Free Fall

When dealing with vertical motion, the acceleration \(a\) is replaced by \(g\).

Crucial Point: Sign Convention!

Since velocity, displacement, and acceleration are vectors, you must pick a consistent direction as positive (e.g., upwards) and stick to it.

• If you choose Upwards as positive:
  • Initial velocity (\(u\)) might be positive (if thrown up).
  • Acceleration (\(a\)) will always be \(-g\) (since gravity pulls down).
• If you choose Downwards as positive:
  • Initial velocity (\(u\)) might be negative (if thrown up).
  • Acceleration (\(a\)) will always be \(+g\) (since gravity pulls down).

Example: If a ball is thrown upwards, when it reaches its highest point, its instantaneous velocity (\(v\)) is zero, but its acceleration is still \(g\) downwards.