Kinematics: Distance, Displacement, Speed, Velocity, Acceleration, SUVAT, Free Fall, Projectile Motion

Distance and displacement

Distance: This is like the total path you walk. If you walk 5 meters forward and then 3 meters back, your distance is $5 + 3 = 8$ meters. It's just a number (scalar). Unit: metre ( $m$ ).
Displacement: This is how far you are from where you started, and in what direction. If you walk 5 meters forward and then 3 meters back, your displacement is $5 - 3 = 2$ meters forward from your start. It has a size and a direction (vector). Unit: metre ( $m$ ).
Circular-track example (A to B): Imagine running on a circular track.
- If you run halfway around (from A to B, directly opposite A), your distance is simply half the track's total length (half the circumference). If the track's radius is 'R', this distance is $(\frac{1}{2}) \times (2 \pi R) = \pi R$ .
- Your displacement from A to B is a straight line from A to B, which is the diameter of the circle ( $2R$ ). It's directed straight across the circle, from A to B.
- If you run all the way back to A (one full lap), your total distance is the entire track length ( $2 \pi R$ ). But your displacement is zero, because you ended up exactly where you started.

Speed and velocity

Speed: How fast you are moving, without caring about direction. It's the rate distance changes. Like your car's speedometer reading. Scalar quantity. Unit: metres per second ( $m/s$ or $m s^{-1}$ ).
Velocity: How fast you are moving and in what direction. It's the rate displacement changes. Vector quantity. Unit: metres per second ( $m/s$ or $m s^{-1}$ ). The sign (positive/negative) tells you the direction.
Relationship between speed and velocity:
- If you drive straight down a road without turning, your speed and the magnitude (size) of your velocity are the same.
- However, if you drive around a bend at a constant speed, your speed doesn't change, but your velocity does change because your direction is changing. Your car's velocity is constantly changing even if its speed is constant while turning.

Constant acceleration and SUVAT (SUVATE) equations

Acceleration: How quickly your velocity changes. If you speed up, slow down, or change direction, you are accelerating. Vector quantity. Unit: metres per second squared ( $m/s^2$ or $m s^{-2}$ ).
Sign convention: We use signs to show direction. If you're moving right or up, it's usually positive. If you're moving left or down, it's negative.
Key equations (for motion in a straight line with constant acceleration): These are often called SUVAT equations, where:
- $s$ = displacement
- $u$ = initial velocity (velocity at the start)
- $v$ = final velocity (velocity at the end)
- $a$ = constant acceleration
- $t$ = time
- $\boxed{v = u + at}$ : This tells you your final velocity if you know your starting velocity, acceleration, and time. (Example: If you start at 0 m/s and accelerate at 2 m/s
  ² for 5 seconds, your final velocity is $0 + 2 \times 5 = 10$ m/s).
- $\boxed{s = ut + \frac{1}{2}at^2}$ : This tells you your displacement given initial velocity, acceleration, and time. (Example: If you start at 0 m/s and accelerate at 2 m/s
  ² for 5 seconds, your displacement is $0 \times 5 + \frac{1}{2} \times 2 \times 5^2 = 25$ meters).
- $\boxed{v^2 = u^2 + 2as}$ : This connects velocities, acceleration, and displacement without needing time.
- $\boxed{s = vt - \frac{1}{2}at^2}$ : Similar to the second, but uses final velocity.
- $\boxed{s = \frac{u+v}{2} \times t}$ : This is useful if you know the average velocity.
Notes:
- These formulas only work if acceleration stays the same and the object moves in a straight line.
- Remember, negative values for $s$ , $u$ , $v$ , or $a$ just mean they are in the opposite direction.
- You can rearrange these equations to find any missing value if you have the others.

Motion graphs

Graphs help us visualize motion.

Displacement-time (s-t) graph:
- Gradient (slope) = Velocity ( $v = \frac{ds}{dt}$ ). A steeper slope means faster velocity.
- Example: If the graph is a straight line going up, you have constant positive velocity. If it's flat, you're not moving (zero velocity).
- The area under this graph doesn't usually mean anything useful in physics.
Velocity-time (v-t) graph: This is very important!
- Gradient (slope) = Acceleration ( $a = \frac{dv}{dt}$ ). A steeper slope means greater acceleration.
- Area under the graph = Displacement (or total distance if you consider positive and negative areas separately).
  - $S = \text{area under } v \text{-}t = \text{displacement}$
- Shape examples:
  - Line going up steadily (positive, constant gradient): You are speeding up at a constant rate (constant positive acceleration).
  - Horizontal line (zero gradient): You are moving at a constant speed/velocity (zero acceleration).
  - Line going down steadily (negative, constant gradient): You are slowing down at a constant rate (constant negative acceleration/deceleration).
  - Line crossing the time axis: When the line crosses from positive velocity to negative velocity, it means the object has momentarily stopped and is now moving in the opposite direction.
Acceleration-time (a-t) graph:
- The area under the graph gives the change in velocity.
- Example: If the graph shows a constant positive acceleration, it means your velocity is increasing steadily over time. If it's a flat line at zero, your velocity is constant.
Sketched behaviour (conceptual):
- If the v-t graph is a straight line sloping up, your speed is increasing steadily.
- If the v-t graph is a flat horizontal line, your speed is steady (not changing).
- If the v-t graph is a straight line sloping down, your speed is decreasing steadily.
- When the v-t graph crosses the horizontal time axis, the object changes its direction of motion.

Exam-style questions: quick solutions (concepts and methods)

Q1: A car’s velocity–time profile shows positive and negative areas.
- Total displacement = The overall "net" area under the v-t graph. Positive area means moving forward, negative area means moving backward. So, add them up considering their signs.
- Average velocity = (Total displacement) ÷ (Total time).
- Example: If you go 30m forward (positive area) and then -30m backward (negative area), your net displacement is $30 + (-30) = 0$ m. So your average velocity for that trip is 0 m/s, even though you moved!
Q2: A mass on a spring bounces vertically (up and down) from t = 0.
- Velocity (speed and direction) changes as it bounces. When the spring stops for a moment at its highest or lowest point, its velocity is zero.
- Its fastest points (maximum velocity) are when it passes through the middle (equilibrium) position.
Q3: Given a velocity–time graph for motion in a straight line, sketch the corresponding displacement–time graph from $t = 0$ to $t = T$ .
- Remember: The slope of the s-t graph tells you the velocity.
- So, if your v-t graph shows increasing velocity, your s-t graph should get steeper. If v-t shows velocity becoming zero, s-t should become flat (zero slope). If v-t shows negative velocity, s-t should be going down.
Q4: A graph shows acceleration vs time for an object starting from rest. Sketch the corresponding velocity–time graph.
- Remember: The area under the a-t graph gives the change in velocity.
- If the acceleration is positive and constant, the area will make the velocity increase steadily from rest (0 m/s). This will look like a straight line going up on the v-t graph. If acceleration is zero, velocity stays constant (horizontal line on v-t graph).

Free fall and gravity

Free fall: This is when an object falls only because of gravity, with no air resistance pushing back (like in a vacuum).
In free fall, all objects accelerate downwards at a constant rate, which we call $g$ . On Earth, $g \approx 9.81 \, \text{m/s}^2$ (or often rounded to $10 \, \text{m/s}^2$ for simpler calculations). This means that for every second an object falls, its downward velocity increases by about $9.81 \, \text{m/s}$ .