Section 3.4 - Motion in Space

Section Objectives

  1. Find the tangential and normal components of acceleration.
  2. Solve a projectile motion problem in space.



Motion vectors

Suppose is a position vector. That is to say that is a twice-differentiable vector-valued function that gives the position of an object in space at time . In this case, the velocity vector and the acceleration vector are given respectively by

Furthermore, the speed is given by

where is the arc-length parameter.



Examples




Projectile motion

Suppose that an object at the point is launched at the angle with an initial speed . If gravity is the only force acting on the object, then it is easy to derive the object's position vector from Newton's 2nd law:

where is acceleration due to the force of gravity.



Examples




Components of the acceleration vector

When an object is accelerating, its velocity is changing. The velocity vector may be changing because its magnitude is changing or its direction is changing (or both). It's intuitively clear then that an accelerating object is either changing speed or direction.


Theorem

Suppose an object is moving along a curve with position vector . Then the acceleration vector lies in the plane determined of the vectors and . Furthermore,

where and .


The scalar functions and are called the tangential and normal components of acceleration, respectively. There are a number of alternative formulas for and :

and



Examples