Motion in 1 dimension

Welcome to the World of Motion!

Welcome to the study of Kinematics! This is the branch of mechanics that describes how things move without worrying yet about what causes that movement (like forces). Whether it is a sprinter on a track or a car braking at a red light, the rules we learn here will help you predict exactly where an object will be and how fast it will be going. Don't worry if this seems a bit abstract at first; we will use plenty of everyday examples to make it click!

1. The Language of Kinematics

To talk about motion, we need to be very specific with our words. In everyday life, we use "speed" and "velocity" to mean the same thing, but in Mathematics, they are quite different!

Scalars vs. Vectors

Scalars are quantities that only have a size (magnitude). Vectors have both a size AND a direction.
• Distance (Scalar): How much ground you covered. Example: "I walked 5 km."
• Displacement (Vector): How far you are from where you started, in a straight line. Example: "I am 3 km North from home."

Speed and Velocity

• Speed is a scalar. It is the distance travelled divided by time.
• Velocity is a vector. It is the rate of change of displacement.
Average Velocity = \( \frac{\text{Overall Displacement}}{\text{Elapsed Time}} \)
Average Speed = \( \frac{\text{Total Distance}}{\text{Elapsed Time}} \)

Memory Tip: Velocity is a Vector (both start with V). Speed is a Scalar (both start with S)!

Acceleration

Acceleration is how quickly velocity is changing. If you speed up, slow down, or change direction, you are accelerating.
• If an object slows down, we often call it deceleration or negative acceleration.

Quick Review:
• Distance: Path taken (always positive).
• Displacement: Straight line from A to B (can be negative).
• Velocity: Speed in a specific direction.

2. Kinematics Graphs

Graphs are a brilliant way to "see" motion. There are two main types you need to master:

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

• The gradient (slope) of the line represents the velocity.
• A flat horizontal line means the object is stationary (velocity = 0).
• A straight diagonal line means constant velocity.
• A curve means the object is accelerating or decelerating.

Velocity-Time (\(v-t\)) Graphs

• The gradient represents the acceleration.
• The area under the graph represents the displacement.
• A flat horizontal line means constant velocity (zero acceleration).
• If the graph goes below the x-axis, the object has changed direction.

Analogy: Think of the gradient as the "steepness" of a hill. The steeper the hill on a displacement graph, the faster you are running!

Key Takeaway: To find velocity from displacement, find the gradient. To find displacement from velocity, find the area.

3. Constant Acceleration (SUVAT)

When an object moves in a straight line with constant acceleration, we can use five special equations. We call these the SUVAT equations after the five variables involved:

s = displacement (m)
u = initial velocity (ms\(^{-1}\))
v = final velocity (ms\(^{-1}\))
a = acceleration (ms\(^{-2}\))
t = time (s)

The SUVAT Equations:

1. \( v = u + at \)
2. \( s = \frac{1}{2}(u + v)t \)
3. \( s = ut + \frac{1}{2}at^2 \)
4. \( s = vt - \frac{1}{2}at^2 \)
5. \( v^2 = u^2 + 2as \)

Did you know? You can derive these! For example, \( v = u + at \) is just the definition of acceleration rearranged: \( a = \frac{v - u}{t} \).

How to Solve SUVAT Problems:

1. Write down your list: Write \(s, u, v, a, t\) and fill in what you know.
2. Identify what you need: Circle the variable the question is asking for.
3. Choose the equation: Find the equation that uses your three knowns and your one unknown.
4. Check your signs: If you decide "Up" is positive, then gravity (\(a\)) must be negative!

Common Mistake: Don't use SUVAT if the acceleration is changing! These only work when a is a constant number.

4. Calculus in Kinematics

What if acceleration isn't constant? This is where Calculus saves the day. We use differentiation and integration to move between displacement, velocity, and acceleration.

The Kinematics Ladder

Think of it as a ladder. To go down the ladder, you differentiate with respect to time (\(t\)). To go up the ladder, you integrate with respect to time.

[Top] Displacement (\(r\) or \(s\))
↓ Differentiate (\( v = \frac{dr}{dt} \))
[Middle] Velocity (\(v\))
↓ Differentiate (\( a = \frac{dv}{dt} = \frac{d^2r}{dt^2} \))
[Bottom] Acceleration (\(a\))

Going Back Up:
• To get Velocity from Acceleration: \( v = \int a \, dt \)
• To get Displacement from Velocity: \( r = \int v \, dt \)

Don't forget: When you integrate, always add the constant of integration (\(+C\))! You usually find \(C\) by using the "initial conditions" (what was happening at \(t = 0\)).

Summary Takeaway:
• Differentiate to find the rate of change (e.g., how fast position is changing).
• Integrate to find the total change (e.g., the total distance moved).

5. Relative Velocity in 1D

In one dimension (a straight line), relative velocity is simply how fast one object looks like it's moving from the perspective of another object.

• If two cars are driving toward each other, they seem to be approaching much faster (you add their speeds).
• If one car is overtaking another, the speed difference seems much smaller (you subtract their speeds).

Example: If Car A is moving at 20 ms\(^{-1}\) and Car B is behind it moving at 25 ms\(^{-1}\), the velocity of B relative to A is \( 25 - 20 = 5 \) ms\(^{-1}\).

Final Encouragement

Kinematics is the foundation of all Mechanics. If you can master the difference between a vector and a scalar, and learn when to use SUVAT versus Calculus, you are well on your way to success! Keep practicing drawing those velocity-time graphs—they are often the "secret key" to solving even the hardest exam questions.