Lecture Notes: Understanding Acceleration

Key Concepts

Velocity vs. Acceleration

• Velocity:

  • A vector quantity (has both magnitude and direction).
  • Describes how fast an object's position is changing over time.
  • Average Velocity is the change in position over time, given by:
    [ V = \frac{\text{Final Position} - \text{Initial Position}}{\text{Elapsed Time}} ]
  • Instantaneous Velocity: As time ( t \to 0 ), the expression gives instantaneous velocity.
  • Example: A car moving at 30 mph east travels a distance of 30 miles every hour, or 60 miles in two hours.

• Speed:

  • Tells how fast distance changes over time.

Acceleration

• Acceleration is the rate at which velocity changes over time.
• Average Acceleration formula:
  [ a = \frac{\Delta v}{t} ]
  • Where ( \Delta v ) is the change in velocity.
  • Instantaneous acceleration is calculated as the time interval approaches zero.
• Example: If acceleration is 8 m/s², velocity changes by 8 m/s each second.

Equations for Motion with Constant Acceleration

• Equation:
  [ v_{final} = v_{initial} + at ]
  • Useful for finding the final speed given initial speed, acceleration, and time.

Problem Examples

Problem 1

• Situation: A car accelerates from 15 m/s to 45 m/s in 5 seconds.
• Solution:
  • Initial Speed ( v_{initial} = 15 \text{ m/s} )
  • Final Speed ( v_{final} = 45 \text{ m/s} )
  • Time ( t = 5 \text{ seconds} )
  • Average Acceleration:
    [ a = \frac{45 - 15}{5} = 6 \text{ m/s}^2 ]

Problem 2

• Situation: A truck accelerates from 25 km/h to 45 km/h in 40 seconds.
• Solution:
  • Initial Speed ( = 25 \text{ km/h} )
  • Final Speed ( = 45 \text{ km/h} )
  • Time ( = 40 \text{ seconds} )
  • Average Acceleration:
    [ a = \frac{20}{40} = 0.5 \text{ km/h/s} ]
  • Convert km/h/s to m/s²:
    [ a = 0.138 \text{ m/s}^2 ]

Problem 3

• Situation: A car accelerates from rest at a rate of 3.5 m/s². Find the speed after 12 seconds.
• Solution:
  • Initial Speed ( = 0 \text{ m/s} )
  • Acceleration ( = 3.5 \text{ m/s}^2 )
  • Time ( = 12 \text{ seconds} )
  • Final Speed:
    [ v = 0 + (3.5)(12) = 42 \text{ m/s} ]

Problem 4

• Situation: A bus accelerates from 12 m/s at 1.2 m/s² for 15 seconds.
• Solution:
  • Initial Speed ( = 12 \text{ m/s} )
  • Acceleration ( = 1.2 \text{ m/s}^2 )
  • Time ( = 15 \text{ seconds} )
  • Final Speed:
    [ v = 12 + (1.2)(15) = 30 \text{ m/s} ]

Problem 5

• Situation: A car at 95 mph brakes to rest in 4 seconds. Calculate acceleration.
• Solution:
  • Initial Speed ( = 95 \text{ mph} = 42.47 \text{ m/s} )
  • Final Speed ( = 0 \text{ m/s} )
  • Average Acceleration:
    [ a = \frac{0 - 42.47}{4} \approx -10.6 \text{ m/s}^2 ]
  • Negative acceleration indicates a decrease in velocity.

Conclusion

• Acceleration: How quickly velocity changes.
• Velocity: How quickly position changes over time.
• Speed vs. Velocity: Speed is scalar; velocity is a vector.