Kinematics Formulas and Motion Types

Kinematics: Types of Motion and Key Formulas

Uniform Motion (MU)

Uniform Motion is characterized by constant velocity, meaning zero acceleration.

Fundamental Equations

• Displacement (D): D = |e_final - e_initial|
• Average Velocity (V_m): V_m = D / t
• Position (e): e = e₀ + v · t

Graphical Representation

• In the Position vs. Time (e/t) graph, the line is straight and inclined.
• In the Velocity vs. Time (V/t) graph, the line is straight and horizontal (zero inclination).

Uniformly Accelerated Motion (MUA)

Uniformly Accelerated Motion is characterized by constant acceleration.

Fundamental Equations

• Acceleration (a): a = (v_final - v_initial) / t
• Velocity (v): v = v₀ + a · t
• Position (e): e = e₀ + v₀t + ¹/₂ · a · t²
• Time-Independent Equation: v² - v₀² = 2 · a · Δe.

  (This equation is used when the variable time (t) does not appear in the problem.)

Graphical Representation

• In the Position vs. Time (e/t) graph, the line must be curved (parabolic).
• In the Velocity vs. Time (V/t) graph, the line must be straight and inclined (showing a slope equal to acceleration).

Movement in Free Fall

Free fall motion is a specific case of MUA where the acceleration is due to gravity (g). We typically use g = 9.8 m/s². The sign convention depends on the chosen reference system.

Types of Vertical Motion

1. Free Fall (Released from Rest)

• Initial Velocity: V₀ = 0
• Acceleration: a = g = 9.8 m/s²
• Final Velocity: V > 0 (if downward is positive)

2. Launch Upward

• Initial Velocity: V₀ > 0
• Acceleration: g = -9.8 m/s² (if upward is positive)
• Upon reaching the maximum height, velocity v = 0.
• Time of Rise (t_rise) equals Time of Fall (t_fall).
• Velocity of Release equals Velocity upon impact with the ground (V_drop).

3. Launch Downward

• Initial Velocity: V₀ > 0
• Acceleration: g = +9.8 m/s² (if downward is positive)
• Final Velocity: V > 0

General Equations for Vertical Motion

Using the sign convention where downward is positive:

• Position: e = e₀ + v₀ · t + ¹/₂ · g · t²
• Final Velocity: v_f = v₀ + g · t

Important Sign Criterion: Time (t) must never be negative.

Unit Conversion

• Kilometers per hour to meters per second: km/h · 1000/3600 = m/s
• Meters per second to kilometers per hour: m/s · 3600/1000 = km/h

Uniform Circular Motion (MCU)

Uniform Circular Motion is motion whose trajectory is circular and covers equal angles in equal times, meaning the angular velocity (ω) is constant.

Angular Units

Angular displacement (Δθ) can be measured in:

• Degrees (º)
• Turns (Revolutions)
• Radians (rad) - SI Unit

Angular velocity (ω) can be measured in:

• º/s
• Turns/s (rps)
• Revolutions per minute (rpm)
• Radians per second (rad/s) - SI Unit

Conversion Factors

1 lap = 1 revolution = 360 degrees = 2π rad

Key Equations for MCU

• Angular Velocity: ω = Δθ / Δt
• Relationship between Linear Velocity (v) and Angular Velocity (ω): v = ω · r (where r is the radius)
• Relationship between Linear Displacement (e) and Angular Displacement (θ): e = θ · r

Acceleration in Circular Motion

Velocity (V) is a vector, characterized by its magnitude, direction, and sense. Acceleration is defined as the change in velocity over time. This change can be caused by a variation in the magnitude or the direction of V.

Intrinsic Components of Acceleration (a)

1. Tangential Acceleration (a_t)

If the magnitude of V varies, there is a tangential acceleration (a_t).

• Direction: Tangent to the trajectory.
• Sense: Same as V if velocity increases, and opposite if velocity decreases.
• Formula: a_t = ΔV / Δt

2. Normal (Centripetal) Acceleration (a_n)

If the direction of V changes, there is a normal acceleration (a_n), which is always directed toward the center of the curve.

• Formula: a_n = V² / r = ω² · r

These two accelerations (a_t and a_n) are called the intrinsic components of the total acceleration (a).

Classification of Motion based on Acceleration Components

• If a_t = 0 and a_n = 0: Uniform Rectilinear Motion (MRU)
• If a_t ≠ 0 and a_n = 0: Uniformly Accelerated Rectilinear Motion (MRUA)
• If a_t = 0 and a_n ≠ 0: Uniform Circular Motion (MCU)

Period and Frequency

Uniform circular motions are periodic motions because the motion is repeated every fixed interval of time.

Period (T)

The time it takes for the object to pass through the same point. In circular motion, it is the time required for one full revolution.

• Formula: T = 1 / f

Frequency (f)

The number of turns (revolutions) the object completes in a unit of time, usually in seconds.

• Formula: f = 1 / T
• Units: 1 / s = s^-1 (Hertz, Hz) - SI Unit