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: $a = \frac{v<em>{f} - v</em>{i}}{t}$ 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:
     $t = \frac{v<em>{f} - v</em>{i}}{a}$

   • 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:
     $v<em>{f} = v</em>{i} + at$

   • 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:
     $v<em>{f} = v</em>{i} + at$

   • 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:
     $t = \frac{v<em>{f} - v</em>{i}}{a}$

   • 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:
     $v<em>{f} = v</em>{i} + at$

   • 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.