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.