Motion in 2D and 3D Space

Motion in Two-Dimensional Space

Motion that occurs in a plane and requires two coordinates to describe position.

Motion in Three-Dimensional Space

Motion that occurs in space and requires three coordinates to describe position.

Position Vector

A vector that extends from the origin of the coordinate system to a point P.

Position Vector Symbol

r

Two-Dimensional Position Vector

r = xî + yĵ

Three-Dimensional Position Vector

r = xî + yĵ + zk̂

Magnitude of a 2D Position Vector

r = √(x² + y²)

Magnitude of a 3D Position Vector

r = √(x² + y² + z²)

Origin

The reference point of the coordinate system.

Coordinate System

A system used to specify the position of a point using coordinates.

Position Coordinates

x, y, and z coordinates specifying the location of a point.

Velocity Vector

The rate of change of the position vector with respect to time.

Velocity Vector Symbol

v

Velocity Vector Formula

v = dr/dt

Two-Dimensional Velocity Vector

v = vxî + vyĵ

Three-Dimensional Velocity Vector

v = vxî + vyĵ + vzk̂

Velocity Components

The horizontal and vertical components of the velocity vector.

Horizontal Velocity Component

vx

Vertical Velocity Component

vy

Acceleration Vector

The rate of change of the velocity vector with respect to time.

Acceleration Vector Symbol

a

Acceleration Vector Formula

a = dv/dt

Two-Dimensional Acceleration Vector

a = axî + ayĵ

Three-Dimensional Acceleration Vector

a = axî + ayĵ + azk̂

Acceleration Components

The horizontal and vertical components of acceleration.

Horizontal Acceleration Component

ax

Vertical Acceleration Component

ay

Projectile Motion

A phenomenon in which an object simultaneously undergoes uniform horizontal velocity and uniform vertical acceleration (free fall).

Projectile

An object given an initial velocity that then follows a path determined entirely by gravitational acceleration.

Trajectory

The path followed by a projectile.

Characteristics of Projectile Motion

Uniform horizontal velocity and uniformly accelerated vertical motion.

Horizontal Motion

The component of projectile motion with constant velocity.

Vertical Motion

The component of projectile motion that follows free-fall motion.

Projectile Motion Assumption

Air resistance is neglected.

Projectile Motion Assumption

Gravity is the only force acting after launch.

Projectile Motion vs. Free Fall

Both experience the same vertical acceleration due to gravity.

Projectile Motion Difference

Projectile motion has horizontal motion, while free fall does not.

Horizontal Velocity in Projectile Motion

Constant throughout the motion.

Horizontal Acceleration in Projectile Motion

Zero.

Vertical Acceleration in Projectile Motion

Constant and equal to −g.

Acceleration Due to Gravity

g = 9.8 m/s² downward.

Independent Motion Principle

Horizontal and vertical motions are independent of each other.

Resultant Motion

The combination of horizontal and vertical motions.

Launch Point

The position where the projectile begins its motion.

Landing Point

The final position of the projectile.

Initial Velocity

The velocity of the projectile at launch.

Initial Velocity Symbol

vi

Launch Angle

The angle between the initial velocity and the horizontal.

Launch Angle Symbol

θi

Initial Velocity Components

The horizontal and vertical components of the initial velocity.

Horizontal Initial Velocity

vix = vi cos θ

Vertical Initial Velocity

viy = vi sin θ

Horizontal Velocity

Constant throughout projectile motion.

Horizontal Velocity Formula

vx = vi cos θ

Vertical Velocity

Changes uniformly due to gravity.

Vertical Velocity Formula

vy = vi sin θ − gt

Horizontal Acceleration

Zero during projectile motion.

Horizontal Acceleration Formula

ax = 0

Vertical Acceleration

Equal to the acceleration due to gravity.

Vertical Acceleration Formula

ay = −g

Horizontal Position

The horizontal location of the projectile at any time.

Horizontal Position Formula

x = xi + (vi cos θ)t

Vertical Position

The vertical location of the projectile at any time.

Vertical Position Formula

y = yi + (vi sin θ)t − ½gt²

Horizontal Displacement

The change in horizontal position.

Horizontal Displacement Formula

Δx = (vi cos θ)t

Vertical Displacement

The change in vertical position.

Vertical Displacement Formula

Δy = (vi sin θ)t − ½gt²

Time of Flight

The total time the projectile remains in the air.

Time of Flight Formula

T = (2vi sin θ)/g

Condition for Time of Flight

The launch and landing heights are the same.

Maximum Height

The greatest vertical position reached by the projectile.

Maximum Height Formula

H = (vi² sin²θ)/(2g)

Condition at Maximum Height

The vertical velocity equals zero.

Vertical Velocity at Maximum Height

0 m/s

Time to Maximum Height

The time required for the projectile to reach its highest point.

Time to Maximum Height Formula

t = (vi sin θ)/g

Horizontal Range

The total horizontal distance travelled by the projectile.

Horizontal Range Formula

R = (vi² sin 2θ)/g

Condition for Horizontal Range

The projectile lands at the same vertical level from which it was launched.

Projectile Trajectory

A parabolic path.

Parabolic Motion

The characteristic path of projectile motion.

Symmetry of Projectile Motion

For equal launch and landing heights, the ascent and descent are symmetrical.

Horizontal Motion Equation

x = vxt

Vertical Motion Equation

y = viyt − ½gt²

Independence of Motion

The horizontal and vertical motions do not affect each other.

Constant Horizontal Motion

Horizontal velocity remains unchanged.

Uniformly Accelerated Vertical Motion

Vertical motion follows the equations of free fall.

Position Vector of a Projectile

r = xî + yĵ

Velocity Vector of a Projectile

v = vxî + vyĵ

Acceleration Vector of a Projectile

a = −gĵ

Horizontal Component of Velocity

Determined using cosine.

Vertical Component of Velocity

Determined using sine.

Resultant Velocity

The vector sum of the horizontal and vertical velocity components.

Magnitude of Velocity

|v| = √(vx² + vy²)

Direction of Velocity

θ = tan⁻¹(vy/vx)

Launch Speed

The magnitude of the initial velocity.

Launch Angle

The angle between the initial velocity and the horizontal axis.

Complementary Launch Angles

Two launch angles that add up to 90°.

Property of Complementary Angles

They produce the same horizontal range if the launch speed is the same.