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. 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:

• Initial Conditions: ,

Example: For an object launched from the origin with velocity , the trajectory is a parabola.

• Time of Flight:

• Range:

• Maximum Height:

Vector Functions: Integration and Initial Conditions

Finding Position from Acceleration

To determine the position function of a particle given its acceleration, initial velocity, and initial position, integrate the acceleration vector and apply the initial conditions.

• Acceleration:

• Initial Velocity:

• Initial Position:

Solution:

• Integrate to get :

• Integrate to get :

Example: The answer is .

Arc Length of Vector Functions

Definition and Calculation

The arc length of a curve parameterized by a vector function is the integral of the magnitude of its derivative over the interval .

• Arc Length Formula:

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

Example: For for :

• 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 from a starting point.

• Arc Length Function:

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

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

Example: For for , the arc length parameterization is for .

Curvature of Vector Functions

Definition and Interpretation

Curvature measures how quickly a curve changes direction. It is defined using the unit tangent vector and the arc length parameter.

• Unit Tangent Vector:

• Curvature:

• For unit speed curves,

Formulas for Curvature:

• Alternative: , where and

Example: For a circle , the curvature is .

Curvature of a Parabola

The curvature of a parabola is highest at the vertex and decreases as increases, meaning the curve flattens out away from the vertex.

• Maximum curvature occurs at (vertex).

• Curvature approaches zero as (curve flattens).

Principal Unit Normal Vector

Definition and Geometric Interpretation

The principal unit normal vector points in the direction in which the curve is turning. It is defined as:

• For other parameters:

At a point , the circle with the same curvature as the curve at has radius and center at .

Additional info: The principal unit normal vector and curvature are fundamental in understanding the geometry of curves in space, especially in applications such as physics and engineering.