1: Motion in 1D (position, velocity, acceleration)

Section 1: Motion in 1D (Position, Velocity, Acceleration)

Motion in one dimension (1D) is the study of objects moving along a straight line. It's the foundation of kinematics and introduces the fundamental quantities describing motion: position, velocity, and acceleration. We define a coordinate system with an origin (O) and a positive direction (e.g., the x-axis). An object's position ($x$) is its location relative to this origin at a specific time ($t$). It's a vector quantity (magnitude and direction implied by sign), measured in meters (m).

The change in position is displacement ($\Delta x$): $\Delta x = x_{\text{final}} - x_{\text{initial}}$. Displacement depends only on the start and end points, not the path taken, and is also a vector. Distance traveled is the total path length (scalar, always positive).

Velocity describes how position changes over time. Average velocity ($v_{\text{avg}}$) is the displacement divided by the time interval: $v_{\text{avg}} = \Delta x / \Delta t$. It gives the overall rate of change. Instantaneous velocity ($v$) is the velocity at an exact moment. It's defined as the derivative of position with respect to time: $v = dx/dt$. Graphically, it's the slope of the position-time ($x$-$t$) graph at a point. Velocity (m/s) is a vector; its sign indicates direction along the line. Speed is the magnitude of velocity ($|v|$), a scalar.

Acceleration describes how velocity changes over time. Average acceleration ($a_{\text{avg}}$) is the change in velocity divided by the time interval: $a_{\text{avg}} = \Delta v / \Delta t$. Instantaneous acceleration ($a$) is the acceleration at an exact instant, defined as the derivative of velocity with respect to time ($a = dv/dt$) or the second derivative of position ($a = d^2x/dt^2$). Graphically, it's the slope of the velocity-time ($v$-$t$) graph. Acceleration (m/s²) is a vector; its sign indicates the direction of the velocity change. A positive acceleration means the velocity is becoming more positive (speeding up if moving positive, slowing down if moving negative).

The relationships between position ($x$), velocity ($v$), and acceleration ($a$) are defined by calculus:

• Velocity is the derivative of position: $v = dx/dt$
• Acceleration is the derivative of velocity: $a = dv/dt = d^2x/dt^2$
• Position is the integral of velocity: $x = \int v dt + x_0$
• Velocity is the integral of acceleration: $v = \int a dt + v_0$

When acceleration is constant ($a = \text{constant}$), these relationships lead to the key kinematic equations (often called SUVAT equations):

1. $v = v_0 + a t$
2. $\Delta x = v_0 t + \frac{1}{2} a t^2$
3. $v^2 = v_0^2 + 2 a \Delta x$
4. $\Delta x = \frac{(v_0 + v)}{2} t$ Where $v_0$ is the initial velocity, $v$ is the final velocity, $\Delta x$ is the displacement, $t$ is the time interval, and $a$ is the constant acceleration. These equations are essential for solving a wide range of 1D motion problems.