Kinematic Equations of Accelerated Motion

Key Concepts

• Instantaneous Velocity: The velocity of an object at a specific instant.
• Instantaneous Acceleration: The rate of change of velocity at a specific instant.
• Equations of Motion (Constant Acceleration):
  • u: Initial velocity
  • v: Final velocity
  • t: Time
  • s: Displacement
  • a: Acceleration (must be constant)
• Calculus Method: Used for deriving equations when acceleration is not constant.
• Average vs. Instantaneous Velocity:
  • Average Velocity: Total displacement divided by total time (Δs/Δt).
  • Instantaneous Velocity: Derivative of displacement with respect to time (ds/dt).
• Velocity-Time Graph: Slope represents instantaneous velocity.
• Displacement from Velocity: For constant velocity, s = vt. For variable velocity, integrate v dt.

Example Calculations

Finding Instantaneous and Average Velocity

• Given: Displacement equation s = 5t² + 4t

• Find: Instantaneous velocity at t = 2s

• Solution:

  • Instantaneous Velocity: v = ds/dt = 10t + 4
  • At t = 2s: v = 24 m/s

• Average Velocity: Between t = 2s and t = 3s

  • Calculate s at t = 3s and t = 2s
  • Use s2 - s1 over t2 - t1
  • Result: 29 m/s

Finding Displacement from Instantaneous Velocity

• Given: Velocity equation v = 10t
• Find: Displacement from 0 to 4s
• Solution:
  • Integrate: s = ∫(0 to 4) 10t dt = 80m

Instantaneous Acceleration

• Formula: a = dv/dt
• Given: Displacement s = 5t² + 4t
• Find: Instantaneous acceleration
• Solution:
  • Find velocity: v = ds/dt = 10t + 4
  • Find acceleration: a = dv/dt = 10 m/s²

Deriving Equations of Motion Using Calculus

First Equation of Motion

• Goal: Derive v = u + at
• Start: a = dv/dt
• Integrate: ∫(u to v) dv = ∫(0 to t) a dt
• Result: v - u = at -> v = u + at

Second Equation of Motion

• Goal: Derive s = ut + 1/2 at²
• Start: v = ds/dt
• Substitute: v = u + at
• Integrate: ds = (u + at)dt
• Result: s = ut + 1/2 at²

Third Equation of Motion

• Goal: Derive v² = u² + 2as
• Start: a = dv/dt
• Eliminate: Time variable via a = v dv/ds
• Integrate: a ∫(0 to s) ds = ∫(u to v) v dv
• Result: v² = u² + 2as

Important Formulas Summary

• Equations for Constant Acceleration:
  1. v = u + at
  2. s = ut + 1/2 at²
  3. v² = u² + 2as
• Instantaneous Equations:
  • Instantaneous Velocity: ds/dt
  • Instantaneous Acceleration: dv/dt
  • Using Integration:
    • Displacement from Velocity: s = ∫v dt
    • Velocity from Acceleration: v = ∫a dt

Practice Question

• Given: Displacement x = 6t + t³
• Find: Acceleration at t = 5s
• Try: Solve and leave answers in comments.

Conclusion

• Practice regularly to master these concepts.
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