AP Physics 1- 1.2 Displacement, Velocity, and Acceleration- Study Notes- New Syllabus
AP Physics 1-1.2 Displacement, Velocity, and Acceleration – Study Notes
AP Physics 1-1.2 Displacement, Velocity, and Acceleration – Study Notes -AP Physics 1 – per latest Syllabus.
Key Concepts:
- Displacement
- Velocity and its type
- Acceleration and its type
Displacement
Displacement is a vector quantity that describes the change in position of an object.
- It is defined as the straight-line distance from the initial position to the final position, along with its direction.
- Notation: \( \Delta \vec{x} = \vec{x}_{f} – \vec{x}_{i} \)
Key points:
- Displacement depends only on the initial and final positions, not on the path taken.
- It can be positive, negative, or zero depending on the chosen coordinate system.
- Displacement can be zero even if the distance traveled is non-zero (e.g., a complete circle).
Example :
A car moves from \( x_i = 2 \, \text{m} \) to \( x_f = 10 \, \text{m} \). Find the displacement.
▶️Answer/Explanation
\( \Delta x = x_f – x_i = 10 – 2 = +8 \, \text{m} \).
Final Answer: Displacement = 8 m to the right.
Example :
A runner runs 400 m around a circular track and returns to the starting point. Find the displacement.
▶️Answer/Explanation
Initial position = Final position.
\( \Delta \vec{x} = \vec{x}_f – \vec{x}_i = 0 \).
Final Answer: Displacement = 0 m, even though distance = 400 m.
Velocity 
Velocity is a vector quantity that describes the rate of change of displacement with respect to time.
Formula:
\( \vec{v} = \dfrac{\Delta \vec{x}}{\Delta t} \) where \( \Delta \vec{x} \) is displacement and \( \Delta t \) is time interval.
Unlike speed (a scalar), velocity specifies both magnitude and direction.
Types of Velocity
Average Velocity:
- Defined over a time interval.
- \( \vec{v}_{avg} = \dfrac{\vec{x}_f – \vec{x}_i}{t_f – t_i} \)
- Represents overall displacement per unit time.
Instantaneous Velocity:
- Velocity at a specific instant of time.
- Defined as the derivative of displacement w.r.t. time:
- \( \vec{v} = \dfrac{d\vec{x}}{dt} \)
- Direction is tangent to the path of motion at that instant.
Uniform Velocity:
- Velocity remains constant in both magnitude and direction.
- Displacement changes equally in equal intervals of time.
Non-uniform (Variable) Velocity:
- Either magnitude or direction (or both) of velocity changes with time.
- Common in accelerated motion or circular motion.
Example :
A car travels 100 m east in 20 s, then 60 m west in 30 s. Find the average velocity over the trip.
▶️Answer/Explanation
Step (1): Net displacement = \( +100 + (-60) = +40 \, \text{m} \) east.
Step (2): Total time = \( 20 + 30 = 50 \, \text{s} \).
Step (3): \( v_{avg} = \dfrac{40}{50} = 0.8 \, \text{m/s} \).
Final Answer: Average velocity = 0.8 m/s east.
Example :
The position of a particle is given by \( x(t) = 4t^2 \, \text{m} \). Find the instantaneous velocity at \( t = 3 \, \text{s} \).
▶️Answer/Explanation
Step (1): Instantaneous velocity = \( v = \dfrac{dx}{dt} \).
Step (2): \( x(t) = 4t^2 \) → \( v(t) = \dfrac{d}{dt}(4t^2) = 8t \).
Step (3): At \( t = 3 \): \( v = 8(3) = 24 \, \text{m/s} \).
Final Answer: Instantaneous velocity = 24 m/s.
Acceleration
Acceleration is a vector quantity that describes the rate of change of velocity with respect to time.
- Formula: \( \vec{a} = \dfrac{\Delta \vec{v}}{\Delta t} \)
- Instantaneous acceleration: \( \vec{a} = \dfrac{d\vec{v}}{dt} \)
Acceleration tells us how quickly velocity changes in magnitude or direction (or both).
Types of Acceleration
Uniform Acceleration:
- Velocity changes by equal amounts in equal intervals of time.
- Example: Free fall under gravity (\( g = 9.8 \, \text{m/s}^2 \)).
Non-uniform (Variable) Acceleration:
- Velocity changes by unequal amounts in equal intervals of time.
- Example: A car speeding up irregularly in traffic.
Positive Acceleration:
- Velocity increases with time.
- Example: A bike accelerating when throttle is increased.
Negative Acceleration (Deceleration or Retardation):
- Velocity decreases with time.
- Example: A car slowing down while brakes are applied.
Average Acceleration:
- Change in velocity over a given time interval.
- \( \vec{a}_{avg} = \dfrac{\vec{v}_f – \vec{v}_i}{t_f – t_i} \)
Example :
A car’s velocity increases from \( 10 \, \text{m/s} \) to \( 30 \, \text{m/s} \) in 10 s. Find the acceleration.
▶️Answer/Explanation
Step (1): Formula → \( a = \dfrac{v_f – v_i}{t} \).
Step (2): Substitute values → \( a = \dfrac{30 – 10}{10} \).
Step (3): \( a = \dfrac{20}{10} = 2 \, \text{m/s}^2 \).
Final Answer: Acceleration = 2 m/s² (positive, speeding up).
Example :
A car slows down from \( 25 \, \text{m/s} \) to \( 5 \, \text{m/s} \) in 4 s. Find the acceleration.
▶️Answer/Explanation
Step (1): Formula → \( a = \dfrac{v_f – v_i}{t} \).
Step (2): Substitute values → \( a = \dfrac{5 – 25}{4} \).
Step (3): \( a = \dfrac{-20}{4} = -5 \, \text{m/s}^2 \).
Final Answer: Acceleration = -5 m/s² (negative, slowing down).