A common use of vector-valued functions is to describe the motion of an object in the plane or in space. A position function
Let
Velocity, denoted
Speed is the magnitude of velocity,
Acceleration, denoted
Watch the video:
Example of Position, Velocity and Acceleration in Three Space from https://youtu.be/gD2R4Jqw6dQ
An object is moving with position function
Find
Sketch
When is the object’s speed minimized?
Solution
Taking derivatives, we find
Note that acceleration is constant.
We can think of acceleration as “pulling” the velocity vector in a certain direction. At
The object’s speed is given by
To find the minimal speed, we could apply calculus techniques (such as set the derivative equal to 0 and solve for
Two objects follow an identical path at different rates on
SolutionWe begin by computing the velocity and acceleration function for each object:
We immediately see that Object 1 has constant acceleration, whereas Object 2 does not.
At
At
In Figure 12.3.2, we see the velocity and acceleration vectors for Object 1 plotted for
Instead, we simply plot the locations of Object 1 and 2 on intervals of
In part (b) of the Figure, we see the points plotted for Object 2. Note the large change in position from
Since the objects begin and end at the same location, they have the same displacement. Since they begin and end at the same time, with the same displacement, they have the same average rate of change (i.e, they have the same average velocity). Since they follow the same path, they have the same distance traveled. Even though these three measurements are the same, the objects obviously travel the path in very different ways.
A young boy whirls a ball, attached to a string, above his head in a counter-clockwise circle. The ball follows a circular path and makes 2 revolutions per second. The string has length 2ft.
Find the position function
Find the acceleration of the ball and give a physical interpretation of it.
A tree stands 10ft in front of the boy. At what
Solution
The ball whirls in a circle. Since the string is 2ft long, the radius of the circle is 2. The position function
(Plot this for
To find
Note how
When the boy releases the string, the string no longer applies a force to the ball, meaning acceleration is
where
An object moves in a helix with position function
SolutionWith
The speed of the object is
The objects in Examples 12.3.3 and 12.3.4 traveled at a constant speed. That is,
There is an intuitive understanding of this. If acceleration is parallel to velocity, then it is only affecting the object’s speed; it does not change the direction of travel. (For example, consider a dropped stone. Acceleration and velocity are parallel — straight down — and the direction of velocity never changes, though speed does increase.) If acceleration is not perpendicular to velocity, then there is some acceleration in the direction of travel, influencing the speed. If speed is constant, then acceleration must be orthogonal to velocity, as it then only affects direction, and not speed.
If an object moves with constant speed, then its velocity and acceleration vectors are orthogonal. That is,
An important application of vector-valued position functions is projectile motion: the motion of objects under only the influence of gravity. We will measure time in seconds, and distances will either be in meters or feet. We will show that we can completely describe the path of such an object knowing its initial position and initial velocity (i.e., where it is and where it is going.)
Suppose an object has initial position
Since the acceleration of the object is known, namely
Knowing
We integrate once more to find
Knowing
(12.3.1) |
We demonstrate how to use this position function in the next two examples.
Sydney shoots her Red Ryder® bb gun across level ground from an elevation of 4ft, where the barrel of the gun makes a
SolutionA direct application of Equation (12.3.1) gives
where we set her initial position to be
(We discarded a negative solution that resulted from our quadratic equation.)
We have found that the bb lands 2.03s after firing; with
Alex holds his sister’s bb gun at a height of 3ft and wants to shoot a target that is 6ft above the ground, 25ft away. At what angle should he hold the gun to hit his target? (We still assume the muzzle velocity is 350ft/s.)
SolutionThe position function for the path of Alex’s bb is
We need to find
This is not trivial (though not “hard”). We start by solving each equation for
Using the Pythagorean Identity
Multiply both sides by
This is a quadratic in
Clearly the negative
Alex has two choices of angle. He can hold the rifle at an angle of about
Consider a driver who sets her cruise-control to 60mph, and travels at this speed for an hour. We can ask:
How far did the driver travel?
How far from her starting position is the driver?
The first is easy to answer: she traveled 60 miles. The second is impossible to answer with the given information. We do not know if she traveled in a straight line, on an oval racetrack, or along a slowly-winding highway.
This highlights an important fact: to compute distance traveled, we need only to know the speed, given by
Let
Note that this is just a restatement of Theorem 12.2.6: arc length is the same as distance traveled, just viewed in a different context.
A particle moves in space with position function
The distance traveled by the particle on
The displacement of the particle on
The particle’s average speed.
Solution
We use Theorem 12.3.1 to establish the integral:
distance traveled | |||
This cannot be solved in terms of elementary functions so we turn to numerical integration, finding the distance to be 12.88 m.
††margin:
(fullscreen)
The displacement is the vector
That is, the particle ends with an
We found above that the particle traveled 12.88 m over 4 seconds. We can compute average speed by dividing: 12.88/4 = 3.22 m/s.
We should also consider Definition 5.4.1 of Section 5.4, which says that the average value of a function
Note how the physical context of a particle traveling gives meaning to a more abstract concept learned earlier.
In Definition 5.4.1 of Chapter 5 we defined the average value of a function
Note how in Example 12.3.7 we computed the average speed as
that is, we just found the average value of
Likewise, given position function
that is, it is the average value of
Let
The average speed is:
The average velocity is:
The next two sections investigate more properties of the graphs of vector-valued functions and we’ll apply these new ideas to what we just learned about motion.
How is velocity different from speed?
What is the difference between displacement and distance traveled?
What is the difference between average velocity and average speed?
Distance traveled is the same as , just viewed in a different context.
Describe a scenario where an object’s average speed is a large number, but the magnitude of the average velocity is not a large number.
Explain why it is not possible to have an average velocity with a large magnitude but a small average speed.
In Exercises 7–10., a position function
In Exercises 11–14., a position function
In Exercises 15–24., a position function
Projectile Motion:
In Exercises 25–28., position functions
Show that the positions are the same at the indicated
Find the velocity, speed and acceleration of the two objects at
In Exercises 29–32., find the position function of an object given its acceleration and initial velocity and position.
In Exercises 33–36., find the displacement, distance traveled, average velocity and average speed of the described object on the given interval.
An object with position function
An object with position function
An object with velocity function
An object with velocity function
Exercises 37–42. ask you to solve a variety of problems based on the principles of projectile motion.
A boy whirls a ball, attached to a 3ft string, above his head in a counter-clockwise circle. The ball makes 2 revolutions per second.
At what
A Cessna flies at 1000ft at 150mph and drops a box of supplies to the professor (and his wife) on an island. Ignoring wind resistance, how far horizontally will the supplies travel before they land?