Lecture on Acceleration and Related Concepts

Key Concepts

Velocity vs. Acceleration

• Velocity: A vector quantity that describes the rate of change of an object's position with respect to time.
  • Formula: ( V = \frac{\Delta \text{Position}}{\Delta \text{Time}} )
  • Instantaneous Velocity: As time ( T ) approaches zero, the expression gives the instantaneous velocity.
  • Example: A car moving at 30 miles/hour east means every hour, the car travels 30 miles.
• Acceleration: A vector quantity describing how fast the velocity is changing with time.
  • Formula: ( A = \frac{\Delta V}{\Delta T} )
  • Instantaneous Acceleration: As ( T ) approaches zero, the expression gives instantaneous acceleration.

Speed vs. Velocity

• Speed: Scalar quantity that tells how fast the distance is changing.
• Velocity: Vector quantity that indicates how fast the displacement is changing.

Equations and Applications

Basic Equations

• Final Velocity: ( V_f = V_i + a \cdot t )
  • Requires constant acceleration.

Example Problems

Problem 1: Car Acceleration

• Given: Initial speed = 15 m/s, Final speed = 45 m/s, Time = 5 seconds.
• Find: Average acceleration.
• Solution:
  • Formula: ( A = \frac{V_f - V_i}{T} )
  • Calculation: ( A = \frac{45 - 15}{5} = 6 ) m/s²

Problem 2: Truck Acceleration

• Given: Initial speed = 25 km/h, Final speed = 45 km/h, Time = 40 seconds.
• Find: Average acceleration in km/h/s and m/s².
• Solution:
  • Acceleration in km/h/s: ( 0.5 ) km/h/s.
  • Conversion to m/s²: ( 0.5 \times \frac{1000}{3600} = 0.1389 ) m/s²

Problem 3: Car from Rest

• Given: Initial speed = 0, Acceleration = 3.5 m/s², Time = 12 seconds.
• Find: Final speed after 12 seconds.
• Solution:
  • Formula: ( V_f = 0 + 3.5 \times 12 = 42 ) m/s

Problem 4: Bus Acceleration

• Given: Initial speed = 12 m/s, Acceleration = 1.2 m/s², Time = 15 seconds.
• Find: Final speed.
• Solution:
  • Formula: ( V_f = 12 + 1.2 \times 15 = 30 ) m/s

Problem 5: Car Deceleration

• Given: Initial speed = 95 miles/hour, Time = 4 seconds.
• Find: Average acceleration in m/s².
• Solution:
  • Convert speed to m/s: ( 42.4 ) m/s.
  • Final speed = 0 m/s (comes to rest).
  • Formula: ( A = \frac{0 - 42.4}{4} = -10.6 ) m/s²
  • Note: Negative acceleration indicates deceleration (slowing down).

Conclusion

• Acceleration provides insight into how the speed changes over time.
• Understanding the difference between speed, velocity, and acceleration is crucial for solving physics problems related to motion.