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How to find the velocity vector given the position vector?
Differentiate the x and y components of the position vector with respect to time: **v(t) = <x'(t), y'(t)>**.
How to find the speed given the velocity vector?
Calculate the magnitude of the velocity vector: **|v(t)| = \sqrt{(x'(t))^2 + (y'(t))^2}**.
How to determine if a particle is moving to the right at a given time?
Evaluate x'(t) at that time. If x'(t) > 0, the particle is moving to the right.
How to determine if a particle is moving upwards at a given time?
Evaluate y'(t) at that time. If y'(t) > 0, the particle is moving upwards.
How to find the position of a particle at a specific time, given the initial position and velocity vector?
Integrate the velocity vector to find the displacement vector, then add the displacement vector to the initial position vector.
How to find the times when the particle is moving only horizontally?
Set y'(t) = 0 and solve for t. This gives the times when the vertical component of velocity is zero.
How to find the times when the particle is moving only vertically?
Set x'(t) = 0 and solve for t. This gives the times when the horizontal component of velocity is zero.
How to calculate the total distance traveled by a particle along the x-axis from t=a to t=b?
Calculate $\int_{a}^{b} |x'(t)| dt$.
How to calculate the total distance traveled by a particle along the y-axis from t=a to t=b?
Calculate $\int_{a}^{b} |y'(t)| dt$.
How to find the acceleration vector given the velocity vector?
Differentiate the x and y components of the velocity vector with respect to time: **a(t) = <x''(t), y''(t)>**
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: $\int |x'(t)| dt$
How to find the total distance traveled along the y-axis?
Integrate the absolute value of the vertical velocity: $\int |y'(t)| dt$
What is the formula to find displacement?
Displacement = $\int_{t_1}^{t_2} v(t) dt = <x(t_2) - x(t_1), y(t_2) - y(t_1)>$
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 = $\int_{a}^{b} |v(t)| dt = \int_{a}^{b} \sqrt{(x'(t))^2 + (y'(t))^2} dt$