Section 4.1: Displacement and Velocity Vectors

Position vector

"A vector from the origin of a chosen coordinate system to the position of a particle in two- or three-dimensional space, represented as r->(t) = x(t)î + y(t)ĵ + z(t)k̂"

Displacement vector

"A vector from the initial position to a final position on a trajectory of a particle, calculated as Δr-> = r->(t₂) - r->(t₁)"

Velocity vector

"A vector that gives the instantaneous speed and direction of a particle; tangent to the trajectory, calculated as v->(t) = dr->(t)/dt"

Instantaneous velocity components

"vₓ(t) = dx(t)/dt, vᵧ(t) = dy(t)/dt, v_z(t) = dz(t)/dt"

Average velocity vector

"v->_avg = Δr->/Δt = [r->(t₂) - r->(t₁)] / (t₂ - t₁)"

Independence of perpendicular motions

"The horizontal and vertical components of motion are independent of each other for a projectile moving in a uniform gravitational field near Earth's surface"

Two-dimensional motion example

"A package dropped from an airplane travels both horizontally and vertically, with its horizontal velocity remaining nearly constant while its vertical velocity changes due to gravity"

Calculating displacement

"To find displacement, subtract the initial position vector from the final position vector"

Finding velocity from position

"To find velocity, take the derivative of the position vector with respect to time"

Vector components

"In three dimensions, vectors can be expressed in terms of î, ĵ, and k̂ unit vectors representing the x, y, and z directions respectively"