Displacement, Velocity & Acceleration

• Displacement ($s$): This is a vector quantity representing the straight-line distance from a fixed origin to the object's current position in a specific direction. Unlike distance, displacement can be positive, negative, or zero, depending on the object's position relative to the starting point.

• Velocity ($v$): Defined as the rate of change of displacement with respect to time, velocity indicates both the speed and the direction of motion. Mathematically, it is the first derivative of the displacement function: $v = \frac{ds}{dt}$.

• Acceleration ($a$): This represents the rate at which velocity changes over time. It is the first derivative of velocity and the second derivative of displacement: $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$.

Finding Motion Variables

• Step 1: Identify the given function: Determine if you are starting with displacement $s(t)$, velocity $v(t)$, or acceleration $a(t)$.
• Step 2: Apply the operator: Differentiate to find rates of change (e.g., $v(t) = s'(t)$) or integrate to find accumulated change (e.g., $s(t) = \int v(t) \, dt$).
• Step 3: Solve for constants: Use boundary conditions, such as 'at $t=0$, the particle is at the origin', to find the value of the integration constant $C$.

Analyzing Specific States

• At Rest: Set the velocity function to zero ($v(t) = 0$) and solve for $t$ to find when the object momentarily stops or changes direction.
• Initial Conditions: Evaluate functions at $t=0$ to find initial displacement, initial velocity, or initial acceleration.

• Displacement vs. Distance: Displacement is the vector change in position ($s_{final} - s_{initial}$), whereas distance is the total scalar path length traveled. To find total distance from a velocity function, you must integrate the absolute value of velocity: $\int |v(t)| \, dt$.

• Velocity vs. Speed: Velocity includes direction (e.g., $-5 \, ms^{-1}$), while speed is the magnitude of velocity (e.g., $5 \, ms^{-1}$). An object can have a changing velocity while maintaining a constant speed if only its direction changes.

• Check the Units: Always ensure units are consistent (e.g., converting $km/h$ to $m/s$ if time is in seconds). Acceleration is typically measured in $ms^{-2}$.

• Identify Keywords: 'Initially' means $t=0$; 'at rest' means $v=0$; 'returning to start' means $s=0$; 'constant velocity' means $a=0$.

• Total Distance vs Displacement: If an exam asks for 'total distance' over an interval where velocity changes sign, you must split the integral at the points where $v(t)=0$ and sum the absolute values of the areas.

• Sanity Check: If an object is 'decelerating', the acceleration vector should be in the opposite direction of the velocity vector. Check if your signs for $a$ and $v$ align with the physical description.

• Forgetting the Constant ($C$): When integrating acceleration to find velocity, or velocity to find displacement, students often forget $+C$. This constant represents the starting state and is crucial for accurate modeling.

• Confusing Average vs Instantaneous: Average velocity is $\frac{\Delta s}{\Delta t}$ over an interval, while instantaneous velocity is the derivative $\frac{ds}{dt}$ at a specific moment. Do not use simple division for functions that are not linear.

• Sign Errors in Direction: A negative acceleration does not always mean 'slowing down'; it means acceleration in the negative direction. If velocity is also negative, the object is actually speeding up in that negative direction.