Kinetics: Displacement, Speed, Velocity, Acceleration

Kinetics Fundamentals

• Kinetics relates to states of motion and how objects move from one place to another.

Displacement and Distance

• Displacement refers to the position of an object relative to its starting point; it is not the same as distance, which is the total length travelled.
• Displacement considers direction; distance does not.
• An aircraft travelling 2 km while veering may have a smaller displacement than the total distance.
• Displacement in an easterly direction would be less than the total displacement.

Speed and Velocity

• Speed and velocity both relate to distance travelled per unit of time.
• Velocity is a vector quantity, thus direction is important.
• Speed is a scalar quantity, thus direction is irrelevant.
• Average speed is the distance travelled divided by the time taken.
• Average velocity is the final displacement divided by the total time.

Acceleration

• Acceleration is when an object's velocity changes over time
• Negative acceleration is called deceleration.
• Average acceleration is the change in velocity divided by the time taken.
• Acceleration is represented by the formula a=ΔV/Δt
• V = fina velocity, u = initial velocit and t = time
• Change in direction, even at a constant speed, is acceleration because acceleration is a vector.

Newton's Laws of Motion

• Newton defined force as energy that produces a change in motion state and formulated three laws of motion.

Newton's First Law

• An object remains at rest, or in motion at a constant speed in a straight line unless acted upon by an unbalanced force
• This concerns inertia, the property of mass that resists changes in motion

Newton's Second Law

• The acceleration of an object relies on the mass and the amount of force applied
• The formula is Force = mass × acceleration (F = ma)
• Friction provides an external force that decelerates an object
• A spacecraft outside the atmosphere accelerates to a new velocity with a single push and maintains it
• With a continuous push from a motor, the spacecraft continues to accelerate
• A special case is W = mg, where g is acceleration caused by Earth's gravity (9.8 m/s^2), which creates the force called weight (W)

Newton's Third Law

• Whenever one object exerts a force on a second object, the second exerts an equal and opposite force on the first
• The upward thrust of a rocket is the reaction to the force propelling the mass of hot gas downward.

Motion and Linear Motion

• Motion is uniform when equal displacements occur in equal time periods, indicating constant velocity.
• Average velocity is the displacement divided by the time.
• Average speed is the distance divided by time.
• The equation for acceleration is a = (V - u) / t, where V is final velocity, u is initial velocity, and t is time.
• For linear motion, distance and displacement are the same, leading to the equations of motion.

Equations of Motion

• Equations of motion are formulas describing a physical system's behavior in motion over time.
• First Equation of Motion: V = u + at
• Second Equation of Motion: s = ut + (1/2)at^2
• Third Equation of Motion: V^2 = u^2 + 2as

Motion Along a Circular Path

• Objects move in a straight path unless acted upon by a force.
• Motion along a circular path is not possible without a force accelerating the object toward the orbit's center
• That force is Centripetal Force.
• Centripetal force accelerates an object towards the circle's center; without it, the object moves linearly.
• The Centrifugal Force is an apparent (pseudo) force acting opposite the centripetal force.
• Newton's Second Law derives centripetal force: F = (mv^2) / r where m = mass, v = velocity, and r = radius
• Doubling the speed requires quadrupling the centripetal force to maintain the radius, important in high-speed applications.

Orbits

• An orbit is a circular or elliptical path of an object around a more massive one with centripetal force from the larger object is the cause
• Space shuttles and satellites orbit Earth based on this principle where orbital speed that is critical.
• If the orbital speed is too high, the satellite escapes; if too low, it plummets to Earth.
• Higher altitudes require slower orbital speeds.
• Planets follow this principle, Mercury has the highest orbital velocity and Neptune the slowest.
• Geosynchronous Orbit occurs at an altitude of 35786 km, where the satellite's speed matches Earth's rotation, useful for communication satellites.
• Astronauts experience "weightlessness" as they are in constant "free-fall", accelerated towards Earth but maintaining orbit.

Periodic Motion and Oscillation

• Periodic motion (simple harmonic motion or SHM) is repetitive motion over a specific time interval, known as oscillation and happens predictably
• Wave energy traces the path of an object in simple harmonic motion.
• Simple harmonic motion occurs around an equilibrium (neutral) point.
• Periodic motion is sinusoidal with a single resonant frequency which allows the system to oscillate at the same rate.
• Oscillations theoretically continue forever, decreasing as energy is lost to friction.
• Key terms for periodic motion are amplitude, time, frequency, and period.
• Amplitude is the maximum distance from equilibrium, measured in meters.
• Period is the time for one repetition/cycle, measured in seconds.
• Frequency is the number of cycles per second, measured in hertz (Hz).
• Period and frequency are reciprocals: T = 1/f or F = 1/T
• Period/frequency does not depend on the oscillation's amplitude.
• Pendulums and springs display periodic motion, dependent on different factors.

Pendulum

• A simple pendulum consists of a mass hanging from a point by a string; one complete cycle takes the same amount of time.
• The restoring force is gravity, bringing the mass to its lowest point.
• The period (T) is proportional to the length (l) of the string and gravitational acceleration (g), measured in seconds
• Mass is not represented in this formula which means the period depends only on length and gravity when gravitational force is constant on Earth.

Mass on a Spring

• Periodic motion includes a mass on a spring that oscillates, where one end is fixed/free to move.
• The restoring force is elasticity of the material, pulling it back to its neutral position.
• The period (T) depends on the mass (m) and spring constant (k): T=2π√(m/k) measured in seconds
• The period for a spring depends only on mass and spring constant, not length or gravitational acceleration.
• The period of a pendulum depends on its length, and is independent of its mass.
• The period of a spring depends on its mass, and is independent of its length.

Vibration

• Vibration is high-frequency periodic motion, common in speakers/guitar strings but the term is not used commonly for low oscillation frequency like pendulums.
• Rotating/reciprocating components in aircraft can cause vibrations that disturb the pilot and can be destructive.
• Vibration results from imbalances and can be reduced with measures like static/dynamic balancing of helicopter rotors
• Vibration worsens over time with wear/impact damage and can cause cyclic loading that leads to component/airframe failure.

Resonance

• Resonance is the the natural/resonant frequency at which an object vibrates
• Two objects with the same natural frequency can vibrate due to wave energy transfer; they are said to be in resonance
• Resonance can cause destructive forces in aircraft, like propellers vibrating at certain engine speeds, leading to fatigue/failure

Harmonics

• Harmonics are multiples of a natural frequency where the first harmonic is the resonant frequency, multiples of that frequency
• Example: If 100Hz is the resonant frequency then the first harmonic is 100 Hz, second is 200 Hz and third 300 Hz
• Harmonics can cause vibrations due to resonance, important to consider in aircraft design.

Mechanical Advantage

• A pulley system can achieve a powerful MA by combining multiple pulleys, reducing the force needed to lift an object by increasing the length of rope needed.
• The MA of a pulley system equals the number of ropes supporting the movable load, not including the rope used.
• The MA of the pulley system illustrated is 4.

Velocity Ratio

• Velocity ratio (VR) is the direct ratio fo two speeds in the same system.
• For a pulley system with a MA of 4, pulling a metre of rope raises the load by 0.25m the rope moves our times as fast as the load is being raised.
• The VR ratio is 4:1
• MA = distance ratio = VR

Mechanical Efficiency

• Efficiency is the ratio of useful work done by the machine and the work applied, measured in joules (J).
• Machines have internal friction, expending energy as heat
• Friction can be reduced by lubricating sliding, pivoting, or rotating parts.
• Levers are efficient due to low internal resistance, minimal energy is lost as heat By contrast, pulley bearing are less so.
• Efficiency is represented as a percentage (work output/work input × 100), or as a dimensionless number (work output/work input).
• A simple machine has an efficiency of less than 100% because of internal friction. A typical simple lever is about 98% efficient, e.g. 49 J/50 J × 100 = 98% Expressed as a dimensionless number, this is 49 J/50 J = 0.98 In this example, from the 50 joules that were applied to the input, 1 joule was wasted by frictional heating at the pivot.