All Flashcards
What does the area under a velocity-time graph represent?
Displacement.
What does the slope of a position-time graph represent?
Velocity.
What does the slope of a velocity-time graph represent?
Acceleration.
How can you determine when an object changes direction from a velocity-time graph?
Look for points where the velocity graph crosses the t-axis (changes sign).
How can you estimate the total distance traveled from a velocity-time graph?
Estimate the area between the curve and the t-axis, considering areas below the axis as positive.
What does a horizontal line on a velocity-time graph indicate?
Constant velocity (zero acceleration).
What does a horizontal line on an acceleration-time graph indicate?
Constant acceleration.
How does the concavity of a position-time graph relate to acceleration?
Concave up indicates positive acceleration; concave down indicates negative acceleration.
What does the area under an acceleration-time graph represent?
Change in velocity.
How can you determine the time intervals when an object is speeding up or slowing down from a velocity-time graph?
Speeding up when velocity and acceleration have the same sign (both positive or both negative); slowing down when they have opposite signs.
Displacement vs. Distance Traveled.
Displacement: Change in position | Distance Traveled: Total path length.
Vector-Valued vs. Parametric Functions.
Vector-Valued: | Parametric:
Velocity vs. Speed.
Velocity: Vector with magnitude and direction | Speed: Magnitude of velocity (scalar).
Differentiation vs. Integration.
Differentiation: Finds rate of change | Integration: Finds accumulation/area.
Position vs. Velocity.
Position: Location at time t | Velocity: Rate of change of position at time t
Velocity vs. Acceleration.
Velocity: Rate of change of position at time t | Acceleration: Rate of change of velocity at time t
Average Velocity vs. Instantaneous Velocity.
Average Velocity: Displacement / Time | Instantaneous Velocity: Velocity at a specific time
Average Speed vs. Instantaneous Speed.
Average Speed: Total Distance / Time | Instantaneous Speed: Speed at a specific time
Displacement with Vector Valued Functions vs. Parametric Functions
Vector Valued: Integrate the vector | Parametric: Integrate each component separately
Distance Traveled with Vector Valued Functions vs. Parametric Functions
Vector Valued: Integrate the magnitude of the derivative of the position vector | Parametric: Integrate the square root of the sum of the squares of the derivatives of x and y with respect to t
Explain the relationship between position, velocity, and acceleration.
Velocity is the derivative of position, and acceleration is the derivative of velocity. Integration reverses this.
Explain the difference between displacement and distance traveled.
Displacement is the change in position; distance traveled is the total path length.
How is arc length related to distance traveled?
Arc length calculates the total length of the path, giving the distance traveled.
What does integrating a velocity function give you?
Integrating a velocity function gives you the displacement.
What does the magnitude of the velocity vector represent?
The magnitude of the velocity vector represents the speed of the object.
How do you find the velocity vector from a position vector?
Differentiate each component of the position vector with respect to time.
How do you find the acceleration vector from a velocity vector?
Differentiate each component of the velocity vector with respect to time.
What does the definite integral of speed represent?
The total distance traveled over the given time interval.
What is the geometric interpretation of displacement?
The area under the velocity-time curve.
Explain how parametric equations describe motion.
Parametric equations describe motion by defining the x and y coordinates as functions of time, .