Study Notes on Acceleration

Quarter 4 Lesson 1: Acceleration

Introduction to Acceleration

  • Definition: Acceleration is the rate at which an object's velocity changes, which can occur in terms of:

    • Magnitude (speed)

    • Direction of motion

  • Types of Acceleration:

    • Constant Acceleration: An object's velocity changes by a constant amount over time.

    • Changing Acceleration: The object’s velocity varies over time.

  • Types of Movement:

    • Linear Acceleration: Movement in a straight line while changing speed.

    • Curvilinear Acceleration: Movement along non-linear paths, e.g., circular motion.

Forces and Acceleration

  • Cause of Acceleration:

    • An unbalanced force acting on an object causes acceleration.

  • Mathematical Expression of Acceleration:

    • The formula for acceleration is: a = \frac{v}{t} where:

      • a = acceleration

      • v = change in velocity

      • t = change in time

    • Interpretation: This equation illustrates the relationship between acceleration and changes in speed.

Positive and Negative Acceleration

  • Positive Acceleration: Occurs when an object speeds up.

  • Negative Acceleration (Deceleration): Occurs when an object slows down.

Examples of Acceleration

  • Common instances of acceleration include:

    • A car accelerating on a flat road at a constant rate.

    • A block sliding on a horizontal surface experiencing a constant force.

  • Types:

    • Uniform Acceleration: Velocity changes at a constant rate.

    • Non-Uniform Acceleration: Velocity changes by non-constant intervals over time.

Understanding Accelerating and Non-Accelerating Objects

Accelerating (A)

  • An object is accelerating if there is a change in velocity, which can manifest as:

    • Change in speed (speeding up or decelerating).

    • Change in direction (turning or moving in a curve).

  • Example: Even if speed remains constant, if the direction changes, the object is still accelerating.

Not Accelerating (NA)

  • An object is not accelerating if:

    • It is at rest (not moving).

    • It moves at a constant speed in a straight line (same speed and same direction).

True or False Assessments

  • Assess statements regarding acceleration by marking True or False:

    • An object accelerates when an unbalanced force is acting on it. (True)

    • Acceleration is defined as the rate of change of an object's velocity. (True)

    • A car moving at a constant speed on a straight road is accelerating. (False)

    • Non-uniform acceleration is characterized by a constant change in velocity over equal time intervals. (False)

    • An example of uniform acceleration is a roller coaster that speeds up and slows down throughout the ride. (False)

Vector vs Scalar Quantities

  • Vector Quantity:

    • Has both magnitude and direction.

    • Examples: acceleration, velocity, weight.

  • Scalar Quantity:

    • Has only magnitude.

    • Examples: speed, mass, volume.

Acceleration Formulas and Examples

Fundamental Formulas

  1. Acceleration Formula: a = \frac{v{f} - v{i}}{t} where:

    • v_{f} = final velocity

    • v_{i} = initial velocity

    • t = time elapsed

Worked Examples

  1. Sample Problem 1: A jeepney changes its velocity from 17 m/s to 23 m/s in 5 seconds.

    • Given:

      • v_{i} = 17 \, \text{m/s}

      • v_{f} = 23 \, \text{m/s}

      • t = 5 \text{ s}

    • Solution:
      a = \frac{23 - 17}{5} = 1.2 \, \text{m/s}^2

    • Answer: The jeepney's acceleration is 1.2 m/s².

  2. Sample Problem 2: A car accelerates at a rate of 3.0 m/s² from an initial velocity of 8.0 m/s to a final velocity of 25.0 m/s.

    • Given:

      • a = 3.0 \, \text{m/s}^2

      • v_{i} = 8.0 \, \text{m/s}

      • v_{f} = 25.0 \, \text{m/s}

    • Required:

      • t = ?

    • Formula:
      t = \frac{v{f} - v{i}}{a}

    • Solution:
      t = \frac{25.0 - 8.0}{3.0} = 5.67 \, \text{s}

    • Answer: It takes 5.67 s for the car to accelerate from 8.0 m/s to 25.0 m/s.

  3. Sample Problem 3: An LRT train accelerates from rest at 1.25 m/s² for 20 seconds.

    • Given:

      • a = 1.25 \, \text{m/s}^2

      • v_{i} = 0 \, \text{m/s} (stops from rest)

      • t = 20 \text{ s}

    • Formula:
      v{f} = v{i} + at

    • Solution:
      v_{f} = 0 + (1.25 & \text{m/s}^2 \times 20 \, \text{s}) = 25 \, \text{m/s}

    • Answer: The final velocity is 25 m/s.

  4. Sample Problem 4: During a race, an athlete accelerates at 1.5 m/s² from an initial speed of 3 m/s for 4 seconds.

    • Given:

      • a = 1.5 \, \text{m/s}^2

      • v_{i} = 3 \, \text{m/s}

      • t = 4 \text{ s}

    • Formula:
      v{f} = v{i} + at

    • Solution:
      v_{f} = 3 + (1.5 \times 4) = 3 + 6 = 9 \, \text{m/s}

    • Answer: The final velocity is 9 m/s.

  5. Sample Problem 5: A pedicab accelerates at 2 m/s² from rest to a final speed of 6 m/s.

    • Given:

      • a = 2 \, \text{m/s}^2

      • v_{i} = 0 \, \text{m/s}

      • v_{f} = 6 \, \text{m/s}

    • Required:

      • t = ?

    • Formula:
      t = \frac{v{f} - v{i}}{a}

    • Solution:
      t = \frac{6 - 0}{2} = 3 \, \text{s}

    • Answer: It takes 3 seconds to reach that final velocity.

  6. Sample Problem 6: A motorcycle traveling at 25 m/s accelerates at 8 m/s² for 5 seconds.

    • Given:

      • v_{i} = 25 \, \text{m/s}

      • a = 8 \, \text{m/s}^2

      • t = 5 \, \text{s}

    • Formula:
      v{f} = v{i} + at

    • Solution:
      v_{f} = 25 + (8 \times 5) = 25 + 40 = 65 \, \text{m/s}

    • Answer: The final speed is 65 m/s.

Conclusion

  • Understanding acceleration involves the grasping of fundamental concepts surrounding forces, motion, and the mathematical interpretations that define changes in velocity.

  • Practical understanding is bolstered through extensive problem-solving and real-world applications.