Calculus, with its two main operations - differentiation and integration - plays an essential role in the study of kinematics, particularly in understanding motion along a straight line. This section delves into how differentiation and integration can be applied to solve problems related to displacement, velocity, and acceleration.

## Application of Differentiation in Kinematics

**Differentiation**helps calculate how fast things change. In kinematics, it's used for finding velocity and acceleration from how position changes over time.**Velocity from Displacement:**- Velocity is how fast position changes.
- It's calculated as the rate of change of displacement $s(t)$ over time $t$: $v(t) = \frac{ds}{dt}$ .

**Acceleration from Velocity:**- Acceleration is how fast velocity changes.
- It's the rate of change of velocity over time, or the second rate of change of displacement: $a(t) = \frac{dv}{dt} = \frac{d^2s}{dt^2}$.

**Integration in Kinematics:**- Integration is the reverse of differentiation. It's used to find displacement from velocity.
- Displacement $s$ is the accumulated total of velocity $v(t)$ over time: $s(t) = \int v(t) \, dt$.

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## Examples

### Problem 1: Calculating Displacement from Velocity

**Question: **Find the displacement of a particle moving in a straight line with velocity $v(t) = 3t^2 - 2t$ from $t = 1$ to $t = 4$ seconds.

**Solution:**

**Understand:**Displacement is the integral of velocity over time.**Set Up Integral:**$s(t) = \int (3t^2 - 2t) \, dt$.**Integration Limits:**From $t = 1$ to $t = 4$.**Integrate:**$\int (3t^2 - 2t) \, dt = t^3 - t^2$ .**Evaluate:**$s = [(4^3 - 4^2) - (1^3 - 1^2)]$ .**Result:**$s = 48$ units.

### Problem 2: Finding Velocity from Acceleration

**Question:** Find the velocity after 5 seconds for a particle starting from rest with acceleration $a(t) = 6t$ .

**Solution:**

**Understand:**Velocity is the integral of acceleration over time.**Initial Condition:**Particle starts from rest, so initial velocity $v(0) = 0$ .**Set Up Integral:**$v(t) = \int 6t \, dt + v(0)$ .**Integrate:**$\int 6t \, dt = 3t^2$ .**Velocity Function:**$v(t) = 3t^2$ .**Find Velocity at**$t=5$**:**$v(5) = 75$ units.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.