Acceleration Notes (Calculus)

Fundamental Relationships  

- Displacement s, velocity v, and acceleration a are all functions of time.  

- Velocity is the rate of change of displacement with respect to time:  

  v = ds/dt 

- Acceleration is the rate of change of velocity with respect to time:  

  a = dv/dt

Using Calculus  

- To move from displacement → velocity → acceleration, use differentiation.  

- To move from acceleration → velocity → displacement, use integration.  

- Each integration introduces a constant of integration (c), which must be determined from initial conditions.  

Velocity–Time Graphs  

- Acceleration is the gradient of the velocity–time graph (found using differentiation).  

- Displacement is the area under the velocity–time graph (found using integration).  

Key Phrases in Questions  

- “Starting from rest” → v = 0 when t = 0.  

- “Initially” → refers to t = 0.  

Integration in Practice  

- Integrating acceleration between two times gives the change in velocity over that interval.  

- Integrating velocity between two times gives the displacement over that interval.  

- Note: Displacement can differ from distance travelled.  

  - To find total distance, calculate areas above and below the x‑axis separately.  

Important Reminders  

- Watch for scalar vs vector keywords:  

  - Scalars: distance, speed, magnitude.  

  - Vectors: displacement, velocity, acceleration.  

- Sketching a velocity–time graph is always helpful:  

  - It shows when the object is stationary.  

  - It helps in calculating total distance travelled clearly.