Welcome to Kinematics!
Welcome to the first part of your journey into Motion. Kinematics is the language we use to describe how objects move—whether it's a pebble falling into a well or a sprinter bursting off the blocks. In this chapter, we don't worry about why things move (that's for the next chapter on Forces); we focus entirely on how they move. Don't worry if the graphs or equations look a bit daunting at first—we'll break them down step-by-step!
1. Describing Motion: The Basics
To describe motion accurately, we need to distinguish between how far something went and where it ended up.
Distance vs. Displacement
- Distance: How much ground an object has covered (a scalar).
- Displacement (\( s \)): The straight-line distance between the start and end point, including the direction (a vector).
Example: If you walk 10m North and then 10m South, your distance is 20m, but your displacement is 0m because you're back where you started!
Speed vs. Velocity
- Average Speed: Total distance divided by total time.
- Instantaneous Speed: The speed at a specific moment in time (like what you see on a car's speedometer).
- Velocity (\( v \)): The rate of change of displacement. It is a vector, meaning it has a direction.
Quick Formula: \( \text{Velocity} = \frac{\Delta \text{Displacement}}{\Delta \text{Time}} \)
Acceleration (\( a \))
Acceleration is the rate at which velocity changes. If you speed up, slow down, or change direction, you are accelerating. It is measured in \( \text{ms}^{-2} \).
Quick Review:
Scalars: Distance, Speed, Time.
Vectors: Displacement, Velocity, Acceleration.
2. Graphical Representations of Motion
Graphs are a physicist’s best friend because they tell a visual story of a journey.
Displacement–Time Graphs
- The gradient (slope) of the line equals the velocity.
- A straight diagonal line means constant velocity.
- A horizontal line means the object is stationary (velocity is zero).
- A curved line means the object is accelerating or decelerating.
Velocity–Time Graphs
- The gradient of the line equals the acceleration.
- The area under the graph equals the displacement.
- Did you know? If the graph is a curve (non-linear), you can estimate the displacement by "counting the squares" under the curve or dividing it into several small trapeziums.
Common Mistake to Avoid: Don't confuse the two! Always check the y-axis label before you start calculating gradients or areas.
Key Takeaway: Gradient of \( s-t \) = Velocity. Gradient of \( v-t \) = Acceleration. Area under \( v-t \) = Displacement.
3. SUVAT: Equations for Constant Acceleration
When an object moves in a straight line with constant acceleration, we use the "SUVAT" equations. These are named after the five variables involved:
- \( s \) = displacement (m)
- \( u \) = initial velocity (\( \text{ms}^{-1} \))
- \( v \) = final velocity (\( \text{ms}^{-1} \))
- \( a \) = constant acceleration (\( \text{ms}^{-2} \))
- \( t \) = time (s)
The Four Equations:
1. \( v = u + at \)
2. \( s = \frac{1}{2}(u + v)t \)
3. \( s = ut + \frac{1}{2}at^2 \)
4. \( v^2 = u^2 + 2as \)
How to solve a SUVAT problem:
1. List what you know: Write down \( s, u, v, a, t \) and fill in the numbers from the question.
2. Identify what you need: Put a question mark next to the variable you are trying to find.
3. Pick the equation: Choose the equation that uses your "knowns" and your "unknown."
4. Rearrange and solve: Plug in the numbers and calculate!
4. Linear Motion and Free Fall
When an object is dropped, it falls because of gravity. If we ignore air resistance, it falls with a constant acceleration called \( g \).
- On Earth, \( g \approx 9.81 \, \text{ms}^{-2} \).
- In free fall problems, the acceleration \( a \) is always \( g \) (usually taken as positive if you define "downwards" as the positive direction).
Determining \( g \) in the lab
You might use a trapdoor and electromagnet arrangement. When the power is cut, the electromagnet releases a ball and a timer starts. When the ball hits the trapdoor, the timer stops. By measuring the height \( s \) and the time \( t \), you can use \( s = ut + \frac{1}{2}at^2 \) (where \( u=0 \)) to find \( g \).
Key Takeaway: Free fall is just a special case of SUVAT where \( a = 9.81 \, \text{ms}^{-2} \).
5. Stopping Distances
This is a real-world application of kinematics that saves lives! The total stopping distance of a car is made of two parts:
Stopping Distance = Thinking Distance + Braking Distance
- Thinking Distance: The distance traveled while the driver reacts. This is influenced by reaction time (tiredness, alcohol, distractions) and the speed of the car. Since speed is constant during this phase, \( s = vt \).
- Braking Distance: The distance traveled while the brakes are applied. This is influenced by the car's speed and the condition of the road/brakes (friction).
Memory Aid: Thinking is about the driver; Braking is about the car and road.
6. Projectile Motion
A projectile is an object thrown or launched into the air (like a football or a rocket). The secret to mastering projectiles is the Golden Rule:
Horizontal and vertical motions are completely independent.
Horizontal Motion:
- Acceleration is zero (if we ignore air resistance).
- Velocity remains constant throughout the flight.
- Use \( \text{distance} = \text{velocity} \times \text{time} \).
Vertical Motion:
- Acceleration is constant: \( a = g = 9.81 \, \text{ms}^{-2} \).
- Velocity changes as the object rises and falls.
- Use SUVAT equations.
Analogy: Imagine two balls. One is dropped straight down, the other is fired horizontally at the same time. They will both hit the ground at the exact same moment because gravity pulls them down at the same rate, regardless of how fast one is moving sideways!
Key Takeaway: To solve projectile problems, split the initial velocity into horizontal and vertical components, then treat them as two separate problems linked only by time (\( t \)).
Quick Review Box:
- Constant Velocity: \( s = vt \)
- Constant Acceleration: Use SUVAT
- Falling: \( a = 9.81 \, \text{ms}^{-2} \)
- Projectiles: Horizontal = Constant velocity; Vertical = Acceleration \( g \).