Welcome to Mechanics: Motion in a Straight Line and Projectiles
Welcome to one of the most fundamental chapters in your AQA Physics course! In this chapter, we explore how things move. Whether it’s a car braking on a motorway or a football being kicked into the air, the rules are the same. By the end of these notes, you will be able to predict exactly where an object will be and how fast it will be going at any moment. Don't worry if the math seems a bit heavy at first—once you see the patterns, it becomes much simpler!
1. Describing Motion: The Basics
Before we jump into the formulas, we need to be clear about our "language." In Physics, we distinguish between how far you've gone and where you actually ended up.
Displacement \( (s) \): This is the straight-line distance between the start and end point, including the direction. It is a vector. Units: meters \( (\text{m}) \).
Velocity \( (v \text{ or } u) \): This is the rate of change of displacement. Instantaneous velocity is the speed and direction at a specific moment. Units: meters per second \( (\text{m s}^{-1}) \).
Acceleration \( (a) \): This is the rate of change of velocity. If an object speeds up, slows down, or changes direction, it is accelerating. Units: meters per second squared \( (\text{m s}^{-2}) \).
2. Motion Graphs
Graphs are a visual way to tell the story of an object's journey. For your exam, you need to know what the gradient (slope) and the area under these graphs represent.
Displacement-Time Graphs \( (s-t) \)
The gradient of an \( s-t \) graph represents the velocity.
- A straight, sloping line means constant velocity.
- A horizontal line means the object is stationary \( (v = 0) \).
- A curved line means the velocity is changing (the object is accelerating).
Velocity-Time Graphs \( (v-t) \)
This is the most common graph you will see in exams because it holds two pieces of information:
- The gradient represents the acceleration.
- The area under the graph represents the displacement \( (s) \).
Quick Tip: If the graph goes below the x-axis, the object has changed direction!
Acceleration-Time Graphs \( (a-t) \)
The area under the graph represents the change in velocity \( (\Delta v) \).
3. Equations of Uniform Acceleration (SUVAT)
When an object moves in a straight line with constant (uniform) acceleration, we use the "SUVAT" equations. These are provided in your data booklet, so you don't need to memorize them, but you must know how to use them!
The variables are:
\( s \) = displacement
\( u \) = initial velocity
\( v \) = final velocity
\( a \) = constant acceleration
\( t \) = time taken
The Equations:
\( v = u + at \)
\( s = \frac{(u + v)}{2}t \)
\( s = ut + \frac{1}{2}at^2 \)
\( v^2 = u^2 + 2as \)
How to solve SUVAT problems:
- Write down "S, U, V, A, T" in a list.
- Fill in the values you know from the question.
- Identify which value you are trying to find.
- Pick the equation that contains your "knowns" and your "unknown."
Common Mistake: Forgetting that these only work if acceleration is constant. If the acceleration changes, you must use a graph or consider the motion in separate stages.
4. Free Fall and Acceleration due to Gravity \( (g) \)
When an object is dropped, it accelerates towards Earth due to gravity. If we ignore air resistance, all objects fall with the same constant acceleration, regardless of their mass.
In your exams, always use: \( g = 9.81 \text{ m s}^{-2} \).
Required Practical 3: Determination of \( g \)
You must know how to determine \( g \) by a free-fall method. Usually, this involves dropping a heavy object (like a steel ball) from a known height \( (s) \) and timing how long it takes to fall \( (t) \).
Using \( s = ut + \frac{1}{2}at^2 \), if the object starts from rest \( (u = 0) \), the equation simplifies to:
\( s = \frac{1}{2}gt^2 \)
If you plot a graph of \( s \) on the y-axis and \( t^2 \) on the x-axis, the gradient will be \( \frac{g}{2} \). Therefore, \( g = \text{gradient} \times 2 \).
5. Real-World Motion: Friction, Lift, and Drag
In the real world, objects don't just fall forever in a vacuum. Other forces act on them.
- Friction: A force that opposes motion between two surfaces.
- Drag (Air Resistance): A force that opposes the motion of an object through a fluid (like air or water). Drag increases as the object's speed increases.
- Lift: An upward force produced by an object (like a wing) moving through a fluid.
Terminal Speed
When an object falls through the air, the drag force increases as it speeds up. Eventually, the drag force equals the weight of the object. At this point:
1. The resultant force is zero.
2. The acceleration is zero.
3. The object moves at a constant speed, called terminal speed.
6. Projectile Motion
A projectile is an object thrown or launched into the air (like a kicked ball). The key to projectiles is independence of motion. We treat the horizontal and vertical components completely separately.
Horizontal Motion (x)
- There is no horizontal acceleration (if we ignore air resistance).
- The horizontal velocity \( (v_x) \) remains constant throughout the flight.
- Equation: \( \text{distance} = \text{velocity} \times \text{time} \)
Vertical Motion (y)
- There is a constant acceleration downwards: \( a = -g = -9.81 \text{ m s}^{-2} \).
- We use the SUVAT equations for all vertical calculations.
- At the very top of the arc, the vertical velocity is zero \( (v_y = 0) \).
Typical Projectile Strategy:
If an object is launched at an angle \( \theta \) with velocity \( U \):
1. Split the velocity into components:
Horizontal: \( u_x = U \cos \theta \)
Vertical: \( u_y = U \sin \theta \)
2. Use the vertical motion to find the time of flight \( (t) \).
3. Use that same time \( (t) \) to find the horizontal range.
Did you know? Because horizontal and vertical motions are independent, if you drop a bullet and fire one horizontally at the same time from the same height, they will both hit the ground at exactly the same moment!
Key Takeaways
- Velocity is the gradient of a displacement-time graph.
- Acceleration is the gradient of a velocity-time graph.
- Displacement is the area under a velocity-time graph.
- SUVAT equations only work when acceleration is constant.
- For projectiles, horizontal velocity is constant, while vertical motion is affected by \( g \).
- Terminal speed is reached when drag equals weight.