Moving Around in Space

When an object is moving in two-dimensional space, we use a two-dimensional vector-valued function to keep track of its position at each time, t.

P(t) = (x(t), y(t))

When an object is moving in three-dimensional space, we use a three-dimensional vector-valued function to keep track of its position at each time, t.

P(t) = (x(t), y(t), z(t))

Example:

The movie at the right shows what happens when a baseball is thrown with an initial velocity of 30 feet per second horizontally and 40 feet per second vertically from an initial position (0, 6). The travels of this baseball can be described by
x(t) = 30 t y(t) = 6 + 40 t - 16t²
or. more compactly, using vector notation by
P(t) = (x(t), y(t))
The movie starts at time t = 0 and runs to time t = 2.6.

Missing animation

Use your CAS wiudow to do the following.

Suppose that a baseball is thrown with an initial velocity of 40 feet per second vertically and 30 feet per second horizontally from an initial height of five feet by a person standing at the origin. Draw a graph showing the travels of this baseball.
Suppose that a baseball is thrown with an initial velocity of 20 meters per second vertically and 10 meters per second horizontally from an initial height of 1.5 meters by a person standing at the origin. Draw a graph showing the travels of this baseball.
Suppose that a car is climbing a circular ramp. The ramp has a radius of 100 feet and climbs 20 feet with each "circle" or "story" of the ramp. Draw a graph showing the car's travels as it climbs three "stories" of the ramp.

The derivative, P'(t), of a vector-valued function P(t) is defined by


                       1
     P'(t) =   Lim    --- (P(t + h) - P(t))
             h --> 0   h

The following theorem shows that we can differentiate a vector-valued function by differentiating each of its coordinates. The proof can be found in Multivariable Calculus, Linear Algebra, and Differential Equations in a Real and Complex World.

Theorem:

P(t) = (p₁(t), p₂(t), ... p_n(t))

Then

P'(t) = (p'₁(t), p'₂(t), ... p'_n(t))

Differentiate each of the following functions and check your answers in your CAS window.

P(t) = (cos t, sin t, t)
G(t) = (40 t, 6 + 30 t - 16t²)
H(t) = (20 t, 40 t, 6 + 30 t - 16t²)

The following theorems are proved in Multivariable Calculus, Linear Algebra, and Differential Equations in a Real and Complex World.

Theorem: (Sum Rule)

(P(t) + G(t))' = P'(t) + G'(t)

Theorem: (Scalar Multiplication Rule)

(c P)'(t) = c P'(t)

Theorem: (Chain Rule)

P(f(t))' = P'(f(t)) f'(t)

Theorem: (Dot Product Rule)

(P(t) . G(t))' = P'(t) . G(t) + P(t) . G'(t)

Definition:

Suppose that an object is traveling in R² or in R³ and its location or position is described by the function f(t). Then its velocity is f'(t); its speed is ||f'(t)||; and its acceleration is f''(t).

Be sure to do the exercises in this section of Multivariable Calculus, Linear Algebra, and Differential Equations in a Real and Complex World.