Using Calculus in 1D

• The Derivative Relationship: Velocity is the first derivative of displacement with respect to time ($v = \frac{ds}{dt}$), and acceleration is the first derivative of velocity ($a = \frac{dv}{dt}$).

• The Integral Relationship: Conversely, velocity is the integral of acceleration ($v = \int a \, dt$), and displacement is the integral of velocity ($s = \int v \, dt$).

• Fundamental Theorem Connection: The change in velocity over an interval $[t_1, t_2]$ is the definite integral of acceleration, while the change in position (displacement) is the definite integral of velocity.

Finding Velocity and Acceleration

• Step 1: Identify the given function (usually displacement $s(t)$).

• Step 2: Differentiate once to find the velocity function $v(t)$.

• Step 3: Differentiate the velocity function to find the acceleration function $a(t)$.

Finding Displacement from Acceleration

• Step 1: Integrate the acceleration function $a(t)$ to find $v(t) = \int a(t) \, dt + C_1$.

• Step 2: Use initial conditions (e.g., velocity at $t=0$) to solve for the constant $C_1$.

• Step 3: Integrate the resulting $v(t)$ function to find $s(t) = \int v(t) \, dt + C_2$.

• Step 4: Use initial position data to solve for the second constant $C_2$.

• Displacement vs. Distance: Displacement is the net change in position, while distance is the total path length traveled. In calculus, displacement is $\int_{t_1}^{t_2} v(t) \, dt$, whereas distance is $\int_{t_1}^{t_2} |v(t)| \, dt$.

• Initial vs. Starting from Rest: 'Initially' refers to the state at $t=0$. 'Starting from rest' specifically implies that the velocity $v=0$ at $t=0$.

• Always Include +C: When performing indefinite integration to find $v(t)$ or $s(t)$, never forget the constant of integration. You must use the provided boundary conditions to find its specific value.

• Check the Variable: Ensure you are integrating or differentiating with respect to time ($t$). If a function is given in terms of another variable, standard kinematic calculus rules may not apply directly.

• Sketch the Graph: If asked for total distance, sketch the velocity-time graph to identify where the velocity crosses the x-axis (where the object changes direction).

• Units Consistency: Always verify that your final units match the derivatives (e.g., if $s$ is in meters, $v$ must be $m/s$ and $a$ must be $m/s^2$).

• Confusing Speed and Velocity: Speed is the magnitude of velocity. Integrating speed gives distance, while integrating velocity gives displacement.

• Ignoring Direction Changes: A common error is calculating displacement when the question asks for distance. If an object moves forward and then backward, the displacement will be less than the distance.

• Misinterpreting 'At Rest': An object is at rest when $v=0$. This does not necessarily mean the acceleration is zero; for example, a ball at the peak of its flight has $v=0$ but $a = -9.8 \, m/s^2$.