Cheat Sheet: Describing Motion Around Us

1. Introduction

1.1 Scope

• Everything in nature is in motion.
• Idealised simplified types of motion: linear, circular, oscillatory.

1.2 Chapter focus

• Study of linear motion (motion in a straight line).
• Study of uniform circular motion.
• Introduction of displacement, average velocity, average acceleration.

2. 4.1 Motion in a Straight Line

2.1 Describing Position

• Reference point (Origin O) specifies position.
• Position = distance and direction from reference point at an instant.
• Instant of time = single clock reading.
• Time interval = duration between two instants.
• Object in motion if position changes with time; at rest if position does not change.
• Direction on a line represented by + and - signs (right positive, left negative).

2.2 Distance and Displacement

2.3 Uniform and Non-uniform Motion

• Uniform motion: equal distances in equal time intervals; constant speed.
• Non-uniform motion: unequal distances in equal time intervals; speed changes.

2.4 Average Speed and Average Velocity

• Magnitude of average velocity equals average speed only for motion in one direction.
• Instantaneous velocity = velocity at an instant as time interval → 0.

2.5 Average Acceleration

• Average acceleration = change in velocity ÷ time interval.
• a = (v - u) / (t₂ - t₁).
• SI unit of acceleration = m s^-2.
• Acceleration is a vector; specify magnitude and direction.
• If speed increases, acceleration is in direction of velocity.
• If speed decreases, acceleration is opposite to velocity (deceleration).
• Acceleration can arise from change in magnitude, change in direction, or both.
• Constant acceleration: equal changes in velocity in equal time intervals.

3. 4.2 Graphical Representation of Motion

3.1 Plotting a Graph

1. Draw two perpendicular axes: x-axis = time, y-axis = position.
2. Choose a suitable scale for each axis.
3. Mark values on axes per chosen scales.
4. Plot each (time, position) point.
5. Join points to obtain line or curve.

3.2 Position-Time Graphs

• Slope of position-time graph = average velocity.
• Average velocity between two instants: v = (s₂ - s₁) / (t₂ - t₁).

3.3 Velocity-Time Graphs

• Slope of velocity-time graph = acceleration; a = (v - u) / (t₂ - t₁).
• Area between velocity-time graph and time axis = displacement.
• For constant velocity: displacement = velocity × time interval.
• For changing velocity: displacement = area of trapezium or triangle under the line.

4. 4.3 Kinematic Equations for Motion in a Straight Line with Constant Acceleration

4.1 Quantities and Meanings

• s = displacement.
• t = time interval.
• u = initial velocity (at t = 0).
• v = final velocity (at time t).
• a = acceleration (constant).

4.2 Kinematic Equations

• These equations are valid only when acceleration is constant.
• For straight-line motion in one direction: distance = magnitude of displacement and speed = magnitude of velocity.
• In two-direction straight-line motion, sign of u, v, a, s indicates direction.

5. 4.4 Motion in a Plane

5.1 Uniform Circular Motion

• Circular motion = motion along a circular path.
• Distance along arc ABC = curve length; displacement = straight-line AC.
• For one revolution: distance = circumference = 2πR.
• For one revolution: displacement = 0.
• Average speed for one revolution: v_av = 2πR / T where T = time for one revolution.
• Average velocity for one revolution = 0.
• Uniform circular motion: motion in a circle with constant speed.
• Key feature: speed constant and direction of velocity changes continuously; velocity is tangent to the circle.
• Consequence: motion is accelerated due to change in direction alone.
• Uniform circular motion is an idealised model.