뒤로Kinematic Equations of Motion: Study Notes
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Kinematic Equations of Motion
Introduction to Kinematics
Kinematics is the branch of physics that describes the motion of objects without considering the causes of motion (forces). The kinematic equations relate displacement, velocity, acceleration, and time for objects moving with constant acceleration.
Displacement: The change in position of an object.
Velocity: The rate of change of position with respect to time.
Acceleration: The rate of change of velocity with respect to time.
Motion with No Acceleration
Constant Velocity Motion
When an object moves with no acceleration, its velocity remains constant. This results in a flat (horizontal) line on a velocity vs. time graph.
Key Equation:
Velocity Change:
Graph: The velocity vs. time graph is a straight, horizontal line (slope = 0).
Example: An object moving at for 3 seconds maintains this velocity throughout.
Position Change with No Acceleration
Uniform Position Change
With constant velocity, the position of an object changes uniformly over time, resulting in a straight line on a position vs. time graph.
Key Equation:
Position Change:
Graph: The position vs. time graph is a straight line with slope equal to the velocity.
Example: An object starting at and moving at reaches after 3 seconds.
Motion with Constant Acceleration
Uniform Velocity Change
When an object moves with constant acceleration, its velocity changes uniformly over time, resulting in a straight (but sloped) line on a velocity vs. time graph.
Key Equation:
Velocity Change:
Graph: The velocity vs. time graph is a straight line with slope equal to the acceleration.
Example: An object with and increases its velocity by every second.
Non-Uniform Position Change
With constant acceleration, the position of an object changes non-uniformly, resulting in a parabolic curve on a position vs. time graph.
Key Equation:
Position Change:
Graph: The position vs. time graph is a parabola, reflecting the increasing rate of position change.
Example: An object starting at with follows a parabolic trajectory in position over time.
Displacement and Area Under Velocity-Time Graphs
Calculating Displacement
The displacement of an object during a time interval equals the area under the trendline on its velocity vs. time graph.
Type of Motion | Displacement Equation | Graphical Area |
|---|---|---|
Uniform Motion (Constant Velocity) | Area of rectangle (height = velocity, base = time) | |
Non-Uniform Motion (Changing Velocity) | Area = rectangle + triangle (sum of areas under the curve) |
Concept Check: Interpreting Graphs
Position and Velocity Graphs
Understanding how position and velocity graphs relate to different types of motion is essential for solving kinematic problems.
Trial Comparison: Different trials may show different position vs. time graphs (e.g., upward or downward slopes).
Velocity Graphs: A flat velocity graph indicates constant velocity; a sloped graph indicates acceleration.
Speeding Up: A velocity graph with a positive slope shows an object continually speeding up.
Free Fall and Acceleration Due to Gravity
Objects in Free Fall
Objects in free fall experience constant acceleration due to gravity, regardless of whether they are moving up or down, provided air resistance is negligible.
Acceleration Due to Gravity:
Direction: Acceleration is always downward, even if the object is moving upward.
Key Points:
At the highest point, velocity is zero but acceleration remains .
On the way up, velocity decreases; on the way down, velocity increases (in the negative direction).
Example: A ball thrown upward slows down until it stops momentarily, then speeds up as it falls back down.
Solving Kinematic Problems
Choosing the Correct Equation
To solve kinematic problems, identify the known and unknown quantities and select the appropriate kinematic equation.
Equations:
Example 1: Finding time to reach the highest point when a ball is thrown upward. Use with .
Example 2: Finding time for a ball to return to its initial height. Use with .
Example 3: Finding displacement when time is unknown. Use .
Average Velocity
Calculating Average Velocity
Average velocity can be calculated as the total displacement divided by the total time, or as the arithmetic mean of the initial and final velocities for constant acceleration.
Equation:
Example: If and , then .
Summary Table: Kinematic Equations
Equation | Variables | Use Case |
|---|---|---|
Final velocity, initial velocity, acceleration, time | Finding velocity after time t | |
Final position, initial position, initial velocity, acceleration, time | Finding position after time t | |
Final velocity, initial velocity, acceleration, displacement | Finding displacement or velocity without time | |
Average velocity, initial and final velocities | Average velocity for constant acceleration |
Additional info: These notes expand on the graphical and conceptual content of the provided lecture slides, including definitions, equations, and examples for clarity and completeness.
