Motion in a Plane (2-D Motion)

Introduction to 2-D Motion

Two-dimensional motion extends the concepts of one-dimensional motion to cases where objects move simultaneously in two perpendicular directions, typically along the x- and y-axes. Each direction is analyzed independently, and time is used to connect the motions.

• Key Point 1: 2-D motion is analyzed by treating the x and y components separately, then combining results using time as a common variable.

• Key Point 2: Position vector:

• Key Point 3: Velocity vector:

• Key Point 4: Acceleration vector:

Velocity in a Plane

Vectors in two dimensions can be described using Cartesian coordinates or in terms of magnitude and direction. The magnitude of a position vector gives the distance from the origin to a point in the plane.

• Key Point 1: The position of a point is given by .

• Key Point 2: The magnitude of the position vector is .

• Example: If a ball is at m, its distance from the origin is m.

Projectile Motion

Fundamentals of Projectile Motion

Projectile motion describes the path of an object launched into the air, moving under the influence of gravity alone. The motion is typically resolved into horizontal and vertical components.

• Key Point 1: The horizontal axis (+x) is the direction of travel; the vertical axis (+y) is "up".

• Key Point 2: The only acceleration is due to gravity, , acting in the direction.

• Key Point 3: Initial conditions are often given as launch speed and launch angle .

• Equations:

  • Horizontal motion:

  • Vertical motion:

  • Where ,

• Example: Noodle drives his box car horizontally off a 50.0 m high cliff with m/s.

  • (a) Time in air:

  • (b) Horizontal distance:

• Example: A football kicker launches a ball at m/s, .

  • (a) Find horizontal range:

  • (b) Find speed before impact: , where ,

Uniform Circular Motion

Characteristics of Circular Motion

Uniform circular motion occurs when an object moves in a circle at constant speed. The direction of velocity changes continuously, but its magnitude remains constant. The acceleration is always directed toward the center of the circle (centripetal acceleration).

• Key Point 1: The distance around a circle is .

• Key Point 2: The period is the time to complete one circle.

• Key Point 3: The speed for one circle is .

• Key Point 4: Centripetal acceleration: .

• Example: Fighter jets flying at m/s, with a maximum tolerable acceleration .

  • Maximum radius:

Circular Motion as a Special Application

When motion in a plane is restricted to a circle, the acceleration is always centripetal, and the velocity vector changes direction but not magnitude. This is a key distinction from linear motion, where acceleration can change both speed and direction.

• Key Point 1: Centripetal acceleration is always perpendicular to velocity and points toward the center.

• Key Point 2: The velocity vector maintains constant magnitude (speed), but its direction changes continuously.

• Example: A car moving at constant speed around a circular track experiences a centripetal acceleration toward the center of the circle.

Worked Examples

Example 1: Horizontal Launch from a Cliff

• Given: m, m/s

• Find: (a) Time in air:

• Find: (b) Horizontal distance:

Example 2: Projectile Launched at an Angle

• Given: m/s,

• Find: (a) Range:

• Find: (b) Final speed:

Example 3: Grasshopper Jump

• Given: m/s, cliff height m, launch angle

• Find: (a) Initial speed components: ,

• Find: (b) Use -direction equations to find time in air, then -direction equations to find horizontal distance.

Summary Table: Key Equations in 2-D Motion

Additional info: These notes expand on the provided slides and images, adding definitions, equations, and context for clarity and completeness. All equations are presented in LaTeX format for academic rigor.