1: Rectilinear motion

Rectilinear Motion

Rectilinear motion describes a particle moving along a straight line. It is the simplest form of motion in kinematics and serves as the foundation for analyzing more complex trajectories. Key parameters include position ($s$), velocity ($v$), and acceleration ($a$), all functions of time ($t$).

Position, Velocity, and Acceleration Relationships

• Position ($s(t)$): Defines the particle’s location on the line.
• Velocity ($v$): The rate of change of position with time. Mathematically, $v = \frac{ds}{dt}$. Velocity indicates both speed and direction (positive/negative for forward/backward motion).
• Acceleration ($a$): The rate of change of velocity with time: $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$. Acceleration can be constant or time-dependent.

Constant Acceleration Equations

For constant $a$, kinematic equations relate $s$, $v$, $a$, and $t$ without calculus:

1. $v = v_0 + at$
2. $s = s_0 + v_0 t + \frac{1}{2}at^2$
3. $v^2 = v_0^2 + 2a(s - s_0)$
   Here, $s_0$ and $v_0$ are initial position and velocity at $t = 0$.

Variable Acceleration: Calculus Approach

When acceleration is not constant, use differentiation and integration:

• Given $s(t)$, find $v(t)$ by differentiating: $v = \frac{ds}{dt}$.
• Given $v(t)$, find $a(t)$ by differentiating: $a = \frac{dv}{dt}$.
• Given $a(t)$, integrate to find velocity: $v = \int a dt + C_1$.
• Given $v(t)$, integrate to find position: $s = \int v dt + C_2$.
  Constants $C_1$ and $C_2$ are determined from initial conditions (e.g., $v(0) = v_0$, $s(0) = s_0$).

Graphical Interpretation

• Velocity: Slope of the $s$-$t$ graph.
• Acceleration: Slope of the $v$-$t$ graph.
• Displacement: Area under the $v$-$t$ graph.
• Velocity change: Area under the $a$-$t$ graph.

Key Problem Types

1. Motion prediction: Given $a(t)$ and initial conditions, derive $v(t)$ and $s(t)$ via integration.
2. Time/distance calculations: Use constant-acceleration equations to solve for unknowns (e.g., stopping distance given initial velocity and deceleration).
3. Graph analysis: Extract motion parameters from $s$-$t$, $v$-$t$, or $a$-$t$ plots.

Example: A particle starts at $s_0 = 2 \text{m}$ with $v_0 = 5 \text{m/s}$, undergoing constant acceleration $a = -3 \text{m/s}^2$. Its velocity at $t = 2 \text{s}$ is:
$v = v_0 + at = 5 + (-3)(2) = -1 \text{m/s}$.
The position is: $s = s_0 + v_0 t + \frac{1}{2}at^2 = 2 + 5(2) + \frac{1}{2}(-3)(2)^2 = 6 \text{m}$.