CH. 5 MOTION - Comprehensive Notes

CH. 5 MOTION

Simple "Menu" of Mechanics

• Mechanics is divided into:
  • Statics: Deals with stationary objects in equilibrium.
  • Moving/Accelerating: Deals with objects in motion.
    • Kinematics: Describes motion (s, v, a, t) without considering the cause.
    • Dynamics: Considers the cause of motion using $F = ma$.
• It's important to familiarize oneself with the "language" of mechanics.

Studying Motions: Instruments for Time

• SI unit for measuring time: second (s)
• Time instant: When an event happens.
• Time intervals: How long an event lasts.
• Stopwatch:
  • Unavoidable time lag due to reaction time.
  • Reaction time ≈ 0.2 s
  • Total reaction time for each measurement = $0.2 + 0.2 = 0.4$ s
  • Not suitable for measuring short time intervals.
• Timer-scaler:
  • Measure relatively short time intervals ($10^{-3}$ s).
  • Start or stop by blocking light or breaking electrical contact.
  • Reaction time = 0

Studying Motions: Instruments for Length

• SI unit for measuring length: metre (m)
• Rule
• Venier caliper:
  • Measure length of very small objects
  • Used for diameters of circular objects, inner diameter of screw nuts, depth of container.
• Micrometer screw gauge:
  • Measure length of very small objects
  • Used for diameters of metal wire or metal ball bearings.

Describing Motion: Distance and Displacement

• Distance: The length of the path an object moves.
• Displacement: The separation between the initial and final positions along a straight line with direction specified.
• SI unit: metre (m)
• Symbol: s
• Example:
  • Distance travelled by a car = 280 m
  • Displacement travelled by the car = 180 m due E

Scalars and Vectors

• Scalars:
  • Physical quantity with only size (magnitude).
  • Examples: Mentioned earlier in the course but not repeated here.
• Vectors:
  • Physical quantity with both size (magnitude) and direction.
  • Examples: Mentioned earlier in the course but not repeated here.

Vectors: Graphical and Mathematical

• Graphical Representation: An arrow from point A to point B represents vector AB.
• Mathematical Notation: $\vec{AB}$ or $\vec{p}$
• $\vec{p}$ is a simpler notation to represent $\vec{AB}$. For example, let x be the distance between Aberdeen and Causeway Bay.
• Vectors are equal if and only if they have the same magnitude and direction.

Negative Vectors

• If vector A points in one direction, then -A is the vector pointing in the opposite direction.

Vectors Operation: Addition and Subtraction

• 1-D Case:
  • Example:
    • Initial position: Shau Kei Wan
    • Final position: Admiralty
    • Displacement = $4.5 + 2 = 6.5$ km due W
  • Example:
    • Initial position: Shau Kei Wan
    • Final position: Causeway Bay
    • Displacement = $6.5 - 2 = 4.5$ km due W
• 2-D Case:
  • Initial position: Shau Kei Wan
  • Final position: Hong Kong Coliseum
  • Magnitude of the displacement, by Pythagoras’ theorem.
  • Direction of the displacement, by trigonometric operation.
  • Magnitude = $\sqrt{4.5^2 + 2.1^2} = 4.97$ km
  • Direction = $tan^{-1}(\frac{4.5}{2.1}) = 65.0°$
  • Displacement = 4.97 km N65.0°W

Tip-to-Tail Method

• To find the vector sum $\vec{p} + \vec{q}$:
  • Join the 'tip' of $\vec{p}$ to the 'tail' of $\vec{q}$.
  • The vector sum is the arrow pointing from the 'tail' of $\vec{p}$ to the 'tip' of $\vec{q}$.
• To find the vector difference $\vec{p} - \vec{q}$:
  • Identify the vectors $\vec{p}$ and $\vec{-q}$.
  • Follow the same steps as addition: $\vec{p} - \vec{q} = \vec{p} + \vec{-q}$ .

Describing Motion: Speed and Velocity

• Speed: Scalar, measures how fast an object moves.
  • SI unit: metre per second (ms-¹) or kilometre per hour (km h-1).
  • $1 \text{ ms}^{-1} = 3.6 \text{ km h}^{-1}$
  • $\text{Average speed} = \frac{\text{Total distance travelled}}{\text{Total time taken}}$
• Velocity: Vector, measures how fast an object moves and the direction of motion.
  • Same unit as speed.
  • Symbol: v
  • $\vec{v} = \frac{\vec{s}}{t}$
  • Direction of average $\vec{v}$ = direction of $\vec{s}$
  • Magnitude of average $\vec{v}$ is not necessarily equal to average speed.
• In 1-D motion, one direction is positive, and the opposite is negative.
  • Example: If eastward is positive, then westward is negative.

Describing Motion: Instantaneous Speed and Velocity

• Instantaneous speed (velocity): speed (velocity) of an object over a very short time interval.

Describing Motion: Uniform Motion

• Uniform motion: An object moves at constant velocity.
  • Constant velocity means:
    • Magnitude of velocity remains unchanged.
    • Direction of velocity remains unchanged.
    • The object MUST MOVE IN A STRAIGHT LINE.
• For uniform motion:
  • Slope of s-t graph = v
  • Slope of v-t graph = a = 0
  • Area under v-t graph = Displacement

Describing Motion: Uniform Accelerated Motion

• Uniformly accelerated motion: An object moves with a constant acceleration.
  • $a = \frac{v - u}{t}$
    • v = final velocity
    • u = initial velocity
    • t = time taken
• For uniformly accelerated motion:
  • Slope of s-t graph = v (changing)
  • Slope of v-t graph = a = non-zero constant
• For an object moves at uniform acceleration, average velocity during the time duration we consider, $\vec{v} = \frac{u + v}{2}$

Types of Motion Graphs

• Displacement-time graph (s-t graph): Describes how the displacement of an object varies with time.
  • y-axis: displacement (m)
  • x-axis: time (s)
• Velocity-time graph (v-t graph): Describes how the velocity of an object varies with time.
  • y-axis: velocity (ms-1)
  • x-axis: time (s)
• Acceleration-time graph (a-t graph): Describes how the acceleration of an object varies with time.
  • y-axis: acceleration (ms-2)
  • x-axis: time (s)

Distance-time Graph and Speed-time Graph

• Ignore the consideration of direction, we will get distance-time graph and speed-time graph.

Studying Motions: Instruments for Motion

• Ticker-tape Timer:
  • A tool to analyze straight line motion (linear motion).
  • Black dots are marked on the tape at regular time intervals.
  • Frequency = 50 Hz
  • Time interval between two dots = $\frac{1}{50} = 0.02$ s = 1 tick
  • The tape is attached to the moving object, as it moves, dots are marked to record the motion.
  • Velocity of the motion increases separation between dots increases.
• Data-logging:
  • Motion sensor: measures position of an object.
  • Data-logger: transfers data to computer.
  • Computer: analyzes and presents data in different forms.

Equations of Motion

• Basic requirement: UNIFORM ACCELERATION!!!
  • t: Time taken
  • a: Acceleration
  • s: Displacement
  • u: Initial velocity
  • v: Final velocity
  • $\vec{v}$: Average velocity
• Equations:
  • $v = u + at$ ------ (1)
  • $s = ut + \frac{1}{2}at^2$ ------ (2)
  • $v^2 = u^2 + 2as$ ------ (3)
• Steps in Applying Equations of Motion:
  • Must only apply to uniformly accelerated motions.
  • Sign convention: Define positive direction, usually choose the initial direction of motion to be positive.
  • Choose suitable equation to find out the solution.

Daily Application: Physics of Stopping a Car

• Reaction time of a driver: time lag between seeing the danger and applying the brake (≈ 0.2 s – 2 s). The reaction time of the driver depend on:
  • The reaction time of the driver.
  • The initial speed of the vehicle .
• Distance travelled during the reaction time: thinking distance.
  • $\text{Thinking distance} = \text{Reaction time} \times \text{Initial Speed}$
• After applying the brake, the distance travelled by the car from this moment until it stops: braking distance.
  • The braking distance depends on:
    • The initial velocity of the car.
    • The braking efficiency of the car.
    • The quality of the tires.
    • The weather and the road condition.
• $\text{Stopping distance} = \text{Thinking distance} + \text{Braking distance}$

Reconstruction of Traffic Accident

• When the driver brakes the car suddenly in an accident, the rubber on the tires wears off, and skid marks are produced on the road.
• The police will investigate the length of skid marks, reconstruct the accident, estimate the speed of the car, and determine whether the driver was speeding.

Vertical Motion under Gravity

• Objects fall to the ground due to gravitational force exerted by the Earth on the objects free falling.
• In a space where air resistance ≈ 0, all objects fall at the same rate.

Acceleration due to Gravity

• Symbol: g
• Direction: always points towards the centre of the Earth.
• Magnitude: 9.81 ms-2
• Object Projecting Downwards: Take downward as positive.
  • u = +3 ms-1
  • a = g = +9.81 ms-2
  • s = +20 m
• Object Projecting Upwards (I): Take upward as positive.
  • u = +10 ms-1
  • a = -g = -9.81 ms-2
  • s = +5.10 m
  • At maximum height, instantaneous velocity = 0.
• Object Projecting Upwards (II): Take upward as positive.
  • u = +10 ms-1
  • v = -2 ms-1
  • a = -g = -9.81 ms-2
  • s = +4.89 m
• Object Projecting Upwards (III): Take upward as positive.
  • u = +10 ms-1
  • v = -12 ms-1
  • a = -g = -9.81 ms-2
  • s = -2.24 m