# Calculus in Kinematics

### Basics of Calculus in Kinematics

**Calculus**plays a vital role in kinematics for understanding the relationships between displacement, velocity, and acceleration.- The process of
**differentiation**helps to examine the rate of change at a specific point in a motion, and is used to shift from displacement to velocity, and velocity to acceleration. **Integration**, the opposite of differentiation, helps to accumulate changes over a time interval, and is used to shift from acceleration to velocity, and velocity to displacement.

### Differentiation Process

- Differentiation is the process of finding the derivative of a function. It represents the rate at which quantities change.
- The
**derivative of a displacement-time graph**gives a velocity-time graph. The slope at any point on the original graph represents the velocity at that time. - Arbitrarily, the
**derivative of a velocity-time graph**gives an acceleration-time graph. The slope at a certain point on the original graph signifies the acceleration at that point. - The terms
**‘gradient’**and**‘slope’**in this context essentially mean the same thing - the rate of change of the function at a certain point.

### Integration Process

- Integration is the reverse operation to differentiation and is indicated by the ∫ symbol.
- In the context of kinematics, integration is used to find the area under a graph which can represent displacement, given a velocity-time graph, or a change in velocity given an acceleration-time graph.
- The
**integral of an acceleration-time graph**gives a velocity-time graph. The total area under the graph between two times gives the change in velocity between those times. - The
**integral of a velocity-time graph**gives a displacement-time graph. Similarly, the total area under the graph between two time points provides the change in displacement between those times. - Understanding how to switch between different types of kinematics graphs using integration and differentiation is a crucial skill in mechanics.

### Calculus Terminology in Kinematics

- The term
**‘constant acceleration’**refers to a situation where the acceleration does not change with time. In such scenarios, kinematic equations can be used to solve problems, and no calculus is required. - The term
**‘non-uniform acceleration’**refers to cases where the acceleration does change with time. Such motions require calculus (either differentiation or integration) to solve problems. - A
**‘function’**in this context is a mathematical statement, usually in the form of an equation, which relates one quantity (e.g., displacement) to another (e.g., time). - The
**‘rate of change’**is a key term in kinematics and calculus. In essence, it refers to how one quantity changes in relation to another over time.