Motion in Space and Vector Calculus

Derivative and Tangent Vector

The derivative of a vector-valued function describes the instantaneous rate of change of position in space and provides the tangent vector to the curve at any point.

• Vector Function: , where , , and are differentiable functions.

• Derivative:

• Tangent Vector: Provided , is the tangent vector at the point .

• Component Form: and

Position, Velocity, Speed, and Acceleration

These quantities describe the motion of an object in three-dimensional space.

• Position:

• Velocity:

• Speed:

• Acceleration:

Indefinite Integral of a Vector-Valued Function

The indefinite integral of a vector function is computed component-wise, resulting in a vector of antiderivatives plus a constant vector.

• General Form: , where and is an arbitrary constant vector.

• Component Form:

Example: Finding Position from Velocity

Given a velocity function and an initial position, the position function can be found by integrating the velocity and applying the initial condition.

• Velocity:

• Initial Position:

• Position Function:

• Method: Integrate each component and solve for constants using the initial condition.

Displacement of a Particle

The displacement of a particle over a time interval is the definite integral of its velocity vector.

• Velocity:

• Displacement:

Curves with Constant Distance from the Origin

Theorem: Motion with Constant

If a curve lies on a sphere (or circle) centered at the origin, its position vector and velocity vector are always orthogonal.

• Statement: If is constant, then .

• Proof Outline: If , then . Differentiating both sides gives , so .

Example: Circle in the Plane

• Curve:

• Velocity:

• Orthogonality:

Example: Sphere in Space

• Curve:

• Distance from Origin: (lies on a sphere of radius 3)

• Orthogonality:

Applications: Newton's Second Law and Projectile Motion

Newton's Second Law

Newton's Second Law relates the acceleration of an object to the sum of forces acting on it.

• Formula:

• Gravity: For gravity,

Example: Thrown Ball in Three Dimensions

• Initial Position:

• Initial Velocity:

• Acceleration:

• Velocity:

• Position:

• Applications: Find time when ball hits ground (-coordinate zero), maximum height (when ), etc.

Two-Dimensional Motion in a Gravitational Field

Projectile motion in a plane under gravity is described by parametric equations for position and velocity.

• Initial Velocity:

• Initial Position:

• Velocity:

• Position:

Summary: Properties of Projectile Trajectory

For an object launched from the origin with initial velocity :

• Time of Flight:

• Range:

• Maximum Height:

Example: Cannonball Trajectory

A cannon is fired at with initial speed $100$ m/s. The range is:

• Range: meters

Additional info: These formulas are derived from integrating the equations of motion under constant acceleration due to gravity, and are fundamental in both calculus and physics applications.