Understanding Kinematics: Speed, Velocity, and Acceleration

Average Speed

The average speed of a moving object is the ratio of the distance traveled along its path to the time taken.

Average speed = Δs / Δt

In vector calculus, the average velocity vector of a moving object is the ratio of its displacement vector to the time taken.

v = Δr / Δt

Instantaneous Velocity

The instantaneous velocity of a body is its velocity at a specific point in its trajectory. Its magnitude is called speed.

Acceleration

Acceleration measures how much the velocity of an object changes per unit time. Since velocity is a vector, changes can affect its magnitude, direction, or both.

Average Acceleration

The average acceleration of an object over a time interval is the change in its velocity divided by the time interval.

a = (v - v₀) / Δt

Instantaneous Acceleration

The instantaneous acceleration of an object at a point in its trajectory is the limit of Δv / Δt as Δt approaches zero.

Classification of Movement

Uniform Rectilinear Motion (URM)

An object undergoes URM when it moves at a constant velocity along a straight path. Its acceleration (a) is zero, and velocity (v) is constant. In this case, the average velocity always equals the instantaneous velocity. The motion can be described by the vector equation:

r = r₀ + v(t - t₀)

URM can be represented by a position-time (s-t) graph, which is always a straight line.

Uniformly Accelerated Rectilinear Motion (UARM)

An object undergoes UARM when it moves with constant acceleration. The average acceleration equals the instantaneous acceleration, which is constant but not zero.

v = v₀ + a(t - t₀)

Circular Motion (CM)

Circular motion occurs when an object moves along a circular path of radius R, where R is constant and not zero.

In CM:

• The magnitude of the position vector remains constant.
• The distance traveled by the object is always a circular arc.
• The velocity vector is always perpendicular to the position vector.

Angular Quantities

• Radians: The radian measure of an angle is the ratio of the corresponding arc length to the radius (R) of the circle.
• Average Angular Velocity (ω): The average angular velocity is the angle rotated by the position vector per unit time, expressed in rad/s.

ω = Δθ / Δt

The relationship between average angular velocity and average linear velocity is:

v = Rω

Uniform Circular Motion (UCM)

UCM occurs when an object moves with constant angular velocity (ω ≠ 0). The instantaneous angular velocity equals the average angular velocity.

• Period (T): The time it takes for an object to complete one revolution (measured in seconds).

T = 2π / ω

• Frequency (f): The number of revolutions per second (measured in Hz).

f = ω / 2π

Uniformly Accelerated Circular Motion (UACM)

UACM occurs when an object moves with constant angular acceleration (α).

α = (ω - ω₀) / (t - t₀)

In UACM, the average angular acceleration equals the instantaneous angular acceleration.

Acceleration in Curvilinear Motion

In curvilinear motion, the velocity vector can change in both magnitude and direction.

• Tangential Acceleration (a_t): Due to the change in the magnitude of velocity. Its direction is tangent to the path.
• Normal Acceleration (a_n): Due to the change in the direction of velocity. It is always present in curvilinear motion. Its direction is perpendicular to the path.

a_n = v² / R