Exploration 4: Kinematics and Integration

Vocabulary flashcards covering kinematics definitions, kinematic equations for constant acceleration, and integral calculus relationships.

Kinematics

The quantitative study of motion (relating position, velocity, and acceleration) of objects without considering the forces involved.

Kinematic Equations Validity Conditions

The conditions requiring an object to be moving in 1-dimension and undergoing a constant acceleration.

Instantaneous Velocity Definition

The rate of change of position with respect to time, defined mathematically as $v = \frac{dx}{dt}$.

Average Velocity Definition

The change in position divided by the change in time, defined mathematically as $v_{\text{avg}} = \frac{\Delta x}{\Delta t}$.

Instantaneous Acceleration Definition

The rate of change of velocity with respect to time, defined mathematically as $a = \frac{dv}{dt}$.

Average Acceleration Definition

The change in velocity divided by the change in time, defined mathematically as $a_{\text{avg}} = \frac{\Delta v}{\Delta t}$.

Kinematic Equation 1 (KE1)

The position-time relationship under constant acceleration: $x = x_i + v_i \Delta t + \frac{1}{2} a (\Delta t)^2$.

Kinematic Equation 2 (KE2)

The velocity-time relationship under constant acceleration: $v = v_0 + a \Delta t$.

Kinematic Equation 3 (KE3)

The velocity-displacement relationship under constant acceleration: $v^2 = v_0^2 + 2 a \Delta x$.

Displacement in 1-Dimension

The change in position of an object moving in one dimension, defined as $\Delta x = x_f - x_i$.

Geometrical Interpretation of Displacement

The physical principle stating that the total area under the velocity vs. time ($v$ vs. $t$) curve equals the total displacement, regardless of the type of motion.

Integration

A calculus operation used to calculate the area under a curve, expressed for displacement as $\Delta x = \int v \, dt$.

Area Under Acceleration vs. Time Curve

The graphical representation equal to the total change in velocity of an object, expressed mathematically as $\Delta v = \int a \, dt$.