Two-Dimensional Motion in a Gravitational Field

Projectile Motion and Vector Functions

Projectile motion describes the path of an object moving in a plane under the influence of gravity, with no other forces acting. The position and velocity of the object are given by vector functions, which allow us to analyze the motion in both horizontal and vertical directions.

• Velocity Function:

• Position Function:

• Trajectory: The path is a segment of a parabola.

Key Properties:

• Time of Flight:

• Range:

• Maximum Height:

Example: An object launched from the origin with initial velocity follows a parabolic trajectory.

Vector Functions: Integration and Initial Conditions

Finding Position from Acceleration

Given a particle's acceleration, initial velocity, and initial position, we can determine its position function by integrating the acceleration and applying initial conditions.

• Acceleration:

• Initial Velocity:

• Initial Position:

Solution Steps:

1. Integrate acceleration to find velocity:

2. Integrate velocity to find position:

Example: The position function for the given initial conditions is .

Arc Length of Vector Functions

Definition and Calculation

The arc length of a curve in space is the total distance traveled along the curve, calculated using the vector function's derivatives.

• Arc Length Formula:

• Interpretation: represents the speed; integrating speed over time gives the distance traveled.

Example: For on :

• Arc length:

Parameterization by Arc Length

Arc Length as a Parameter

Parameterizing a curve by arc length means re-expressing the curve so that the parameter directly measures the distance along the curve. This is useful for ensuring constant speed and simplifying calculations.

• Arc Length Function:

• Unit Speed: If , then is the arc length parameter.

Example: The unit circle is parameterized by arc length since .

Example: For on , the arc length parameterization is with .

Curvature of Vector Functions

Definition and Interpretation

Curvature measures how quickly a curve changes direction. It is a fundamental concept in understanding the geometry of curves in space.

• Unit Tangent Vector:

• Curvature (arc length parameter):

• Curvature (general parameter):

• Alternative Formula:

Example: For a circle , the curvature is .

Interpretation: Larger radius means smaller curvature; smaller radius means larger curvature.

Curvature of a Parabola

The curvature of a parabola varies along the curve. It is greatest at the vertex and decreases as the curve flattens out.

• Maximum Curvature: At the vertex of the parabola.

• Curve Flattens: As , curvature .

Example: The diagram shows the parabola and its curvature function, illustrating how curvature is highest at the vertex and decreases as the curve flattens.

Principal Unit Normal Vector

Definition and Geometric Meaning

The principal unit normal vector points in the direction in which the curve is turning. It is orthogonal to the tangent vector and is used to define the center of curvature.

• Arc Length Parameter:

• General Parameter:

• Center of Curvature: At point , the center is

Example: At any point on a curve, the circle with radius and center at matches the curve's curvature and tangent direction at that point.

Additional info: The principal unit normal vector is not testable material but is included for conceptual understanding.