Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. It focuses on concepts such as displacement, velocity, acceleration, and time.

Kinematics provides the mathematical tools to describe and analyze various types of motion, including:

• Linear motion (motion along a straight line)
• Projectile motion (motion under the influence of gravity only)
• Circular motion (motion along a circular path)

Position refers to the location of an object relative to a reference point or origin. It is typically represented by coordinates in a chosen coordinate system.

Displacement is the change in position of an object. It is a vector quantity, having both magnitude and direction. The displacement vector points from the initial position to the final position.

Mathematically, displacement (Δx) is given by:

Δx = x_f - x_i

Where x_f is the final position and x_i is the initial position.

Distance is the total length of the path traveled by an object. Unlike displacement, distance is a scalar quantity (it has magnitude but no direction) and is always positive.

For example, if you walk 3 meters east, then 4 meters north, your:

• Distance traveled = 3 m + 4 m = 7 m
• Displacement = 5 m at an angle of about 53° north of east (calculated using the Pythagorean theorem)

Velocity is the rate of change of displacement with respect to time. It is a vector quantity, having both magnitude and direction.

Average velocity (v_avg) is given by:

v_avg = Δx/Δt

Where Δx is the displacement and Δt is the time interval.

Instantaneous velocity (v) is the velocity at a specific moment in time, defined as the limit of the average velocity as the time interval approaches zero:

v = lim_Δt→0 Δx/Δt = dx/dt

Speed is the rate of change of distance with respect to time. It is a scalar quantity (magnitude only) and is always positive.

Average speed is given by:

Average speed = Total distance / Total time

Instantaneous speed is the magnitude of the instantaneous velocity.

Acceleration is the rate of change of velocity with respect to time. It is a vector quantity, having both magnitude and direction.

Average acceleration (a_avg) is given by:

a_avg = Δv/Δt

Where Δv is the change in velocity and Δt is the time interval.

Instantaneous acceleration (a) is the acceleration at a specific moment in time, defined as the limit of the average acceleration as the time interval approaches zero:

a = lim_Δt→0 Δv/Δt = dv/dt

An object can accelerate by changing its speed, its direction, or both. For example:

• A car speeding up in a straight line has acceleration in the same direction as its velocity
• A car slowing down in a straight line has acceleration in the opposite direction to its velocity
• A car moving at constant speed around a curve has acceleration perpendicular to its velocity, toward the center of the curve

For motion with constant acceleration in a straight line, we can use a set of equations known as the kinematic equations or equations of motion:

v = v₀ + at

x = x₀ + v₀t + ½at²

v² = v₀² + 2a(x - x₀)

x = x₀ + ½(v₀ + v)t

Where:

• x₀ is the initial position
• x is the final position
• v₀ is the initial velocity
• v is the final velocity
• a is the constant acceleration
• t is the time elapsed

These equations can be used to solve problems involving motion with constant acceleration, such as objects falling under gravity (where a = g ≈ 9.8 m/s² downward).

Projectile motion is the motion of an object thrown or projected into the air, subject only to the acceleration of gravity. The path followed by a projectile is called its trajectory, which is a parabola in the absence of air resistance.

Key characteristics of projectile motion:

• The horizontal and vertical components of motion can be analyzed independently
• The horizontal velocity remains constant (assuming no air resistance)
• The vertical motion is subject to constant acceleration due to gravity

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

Initial horizontal velocity: v_0x = v₀cos(θ)

Initial vertical velocity: v_0y = v₀sin(θ)

Horizontal position at time t: x = v_0xt

Vertical position at time t: y = v_0yt - ½gt²

For a projectile launched from ground level (y₀ = 0) and returning to ground level:

• Time of flight: T = 2v_0y/g = 2v₀sin(θ)/g
• Range (horizontal distance): R = v_0xT = v₀²sin(2θ)/g
• Maximum height: H = v_0y²/(2g) = v₀²sin²(θ)/(2g)

The range is maximum when θ = 45°.