Section Objectives
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.
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.
A cannonball is fired from a cannon on a cliff toward the sea. The cannon is aimed at an angle of above horizontal and the initial speed of the cannonball is 60 ft/sec. The cliff is 100 feet above the water.
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