All Flashcards
Explain the significance of the sign of x'(t).
The sign of x'(t) indicates the direction of horizontal movement: positive means right, negative means left.
Explain the significance of the sign of y'(t).
The sign of y'(t) indicates the direction of vertical movement: positive means up, negative means down.
How is speed related to velocity?
Speed is the magnitude (or absolute value) of the velocity vector; it represents how fast the particle is moving, regardless of direction.
What does the derivative of the velocity vector represent?
The derivative of the velocity vector is the acceleration vector, which describes the rate of change of velocity.
Describe the relationship between position, velocity, and acceleration.
Velocity is the derivative of position, and acceleration is the derivative of velocity. Integration reverses this relationship.
How do you determine when a particle is at rest?
A particle is at rest when both the horizontal and vertical components of its velocity are zero, i.e., x'(t) = 0 and y'(t) = 0.
What is the physical meaning of integrating the velocity vector?
Integrating the velocity vector gives the displacement vector, representing the change in position of the particle.
Explain the concept of arc length in the context of vector-valued functions.
Arc length represents the total distance traveled by the particle along its path over a given time interval. It's found by integrating the speed.
How can you find the times when a particle changes direction?
A particle changes direction when either x'(t) or y'(t) changes sign. Solve for when x'(t) = 0 or y'(t) = 0 and check for sign changes.
Explain the difference between displacement and total distance traveled.
Displacement is the change in position from start to end, while total distance traveled considers the entire path, including any changes in direction.
Define position vector.
A vector, p(t), that gives the location of a particle at time t.
Define velocity vector.
A vector, v(t), that describes the rate of change of position with respect to time.
What is speed?
The magnitude of the velocity vector, |v(t)|, representing the rate of travel irrespective of direction.
What does the magnitude of the position vector represent?
The distance of the particle from the origin at time t, denoted as |p(t)|.
What is meant by horizontal velocity?
The x-component of the velocity vector, x'(t), indicating movement along the x-axis.
What is meant by vertical velocity?
The y-component of the velocity vector, y'(t), indicating movement along the y-axis.
What does x'(t) > 0 imply?
The particle is moving to the right.
What does y'(t) < 0 imply?
The particle is moving downwards.
What is a vector-valued function?
A function that maps a real number (usually time) to a vector.
What is the relationship between position and velocity?
Velocity is the derivative of position with respect to time.
Formula for position vector p(t).
p(t) = <x(t), y(t)> = x(t)i + y(t)j
Formula for velocity vector v(t).
v(t) = <x'(t), y'(t)>
Formula for speed.
|v(t)| = \sqrt{(x'(t))^2 + (y'(t))^2}
How to find the total distance traveled along the x-axis?
Integrate the absolute value of the horizontal velocity:
How to find the total distance traveled along the y-axis?
Integrate the absolute value of the vertical velocity:
What is the formula to find displacement?
Displacement =
What is the formula for acceleration vector a(t)?
a(t) = <x''(t), y''(t)>
How to find the unit tangent vector T(t)?
T(t) = v(t) / |v(t)|
How to find the unit normal vector N(t)?
N(t) = T'(t) / |T'(t)|
How to find the arc length of a curve defined by a vector-valued function?
Arc Length =