OSAT Physics (114)

Kinematics is the branch of mechanics that describes motion without considering its causes. It provides the mathematical framework for analyzing how objects move through space and time.

Displacement is a vector quantity representing the change in position of an object. Unlike distance, which measures the total path length traveled, displacement only considers the straight-line distance between initial and final positions along with direction. If an object moves from position x₁ to position x₂, the displacement is Δx = x₂ - x₁.

Velocity describes the rate of change of displacement with respect to time. Average velocity equals total displacement divided by total time elapsed: v_avg = Δx/Δt. Instantaneous velocity is the velocity at a specific moment, found by taking the limit as the time interval approaches zero. Speed is the scalar magnitude of velocity and is always positive.

Acceleration measures the rate of change of velocity with respect to time. Average acceleration is a_avg = Δv/Δt. Positive acceleration in the direction of motion increases speed, while negative acceleration (deceleration) decreases speed. An object can have zero velocity but nonzero acceleration, such as a ball at the peak of its trajectory.

When acceleration remains constant, four kinematic equations describe the motion. Pick the one that leaves out the quantity you neither know nor need:

These equations relate position (x), initial position (x₀), velocity (v), initial velocity (v₀), acceleration (a), and time (t). Problem-solving requires identifying known quantities and selecting the appropriate equation.

Free fall occurs when gravity is the only force acting on an object. Near Earth's surface, the acceleration due to gravity is approximately g = 9.8 m/s² directed downward. Air resistance is neglected in ideal free-fall problems.

For objects dropped from rest, initial velocity is zero. For objects thrown upward, velocity decreases until reaching zero at maximum height, then the object falls back down. The time to reach maximum height equals the time to fall back to the starting point. At any given height, the speed during ascent equals the speed during descent.

Projectile motion combines horizontal motion with constant velocity and vertical motion with constant acceleration due to gravity. These two components are independent and can be analyzed separately.

For a projectile launched with initial velocity v₀ at angle θ above horizontal:

The trajectory forms a parabola. Maximum range occurs at a launch angle of 45° when landing at the same height. Two complementary angles (such as 30° and 60°) produce the same range but different maximum heights and flight times.

Vectors have both magnitude and direction. Vector addition follows the head-to-tail method or component method. To add vectors using components, resolve each vector into x and y components, add corresponding components, then find the resultant magnitude and direction.

For vector A with magnitude A at angle θ:

Aₓ = A cos θ

Aᵧ = A sin θ

Resultant magnitude: R = √(Rₓ² + Rᵧ²) Resultant direction: θ = tan⁻¹(Rᵧ/Rₓ)

Relative velocity problems require vector addition. If object A moves relative to object B, and B moves relative to the ground, then A's velocity relative to the ground is the vector sum of these velocities.

An object moving in a circle at constant speed experiences uniform circular motion. Although speed is constant, velocity continuously changes direction, resulting in acceleration directed toward the center of the circle. This centripetal acceleration has magnitude:

a_c = v²/r = ω²r

where v is the tangential speed, r is the radius, and ω is the angular velocity in radians per second.

The period T is the time for one complete revolution. The frequency f is the number of revolutions per unit time. These are related by T = 1/f. Tangential speed relates to period by v = 2πr/T.

Rotational motion uses angular quantities analogous to linear quantities. Angular displacement (θ) corresponds to linear displacement. Angular velocity (ω = dθ/dt) corresponds to linear velocity. Angular acceleration (α = dω/dt) corresponds to linear acceleration.

For constant angular acceleration, the kinematic equations become:

ω = ω₀ + αt

θ = θ₀ + ω₀t + ½αt²

ω² = ω₀² + 2α(θ - θ₀)

Linear and angular quantities relate through the radius: s = rθ, v = rω, a_tangential = rα. A point on a rotating object also has centripetal acceleration a_c = ω²r = v²/r.

Torque measures how effectively a force causes rotation: τ = rF sin θ, where r is the distance from the pivot to the point where the force acts and θ is the angle between r and F. Only the component of force perpendicular to the lever arm produces torque. Newton's second law for rotation is τ_net = Iα, where I is the moment of inertia.

Moment of inertia depends on how mass is distributed about the axis: a thin hoop has I = MR², a solid cylinder or disk ½MR², and a solid sphere ⅖MR². The more mass far from the axis, the larger I and the harder the object is to spin up. That is why, rolling from rest down the same incline, a solid sphere beats a solid cylinder, which beats a hoop.

Example: A uniform 4.0 m beam weighing 200 N rests on supports at each end, and a 600 N load hangs 1.0 m from the left end. Taking torques about the right support: F_left(4.0) = 200(2.0) + 600(3.0), so F_left = 550 N, and F_right = 800 − 550 = 250 N.

Angular momentum L = Iω is conserved when no external torque acts. A spinning skater who pulls in her arms lowers I, so ω increases in proportion.