Calculus originated as a mathematical way of studying change. Motion is the most natural sort of change, so it makes sense to start our journey there.
In 1D, we can find physical quantities like velocity and acceleration through differentiation of the position along the -axis at time Life becomes more interesting when things move in more than one dimension.
In we need functions (representing the coordinates of the moving object) bundled together into an ordered list called a vector. Velocity and acceleration are tied to the derivatives of these functions, as we'll see.
Beginning here shows us why melding vectors with calculus is so useful. No prior physics experience is needed, but an interest in mechanics would make this quiz much more enjoyable!
We first apply vectors in multivariable calculus to the problem of describing a line in Assuming goes through and is parallel to we found for the position vectors of points on the line.
If we think of as time, (the red arrow below with base at the origin and tip on the line) describes the position of an object moving along
In the visualization above, you have control over the direction of via angles and , 's length and the point
Experiment with the sliders and determine what quantity is most closely associated with the object's speed along the line.
For linear motion in we can explicitly find and The rate of change of the object's motion in the -direction is similar statements hold for and Bundling these derivatives together into the vector gives us complete information about the velocity of the object through space.
For example, we'll learn in Vector-valued Functions that the speed is just the magnitude of velocity
If an object moves along the line compute its speed and enter your answer as an integer on the right.
Similar logic applies when an object travels along a line with non-constant speed. In the visualization below, you once more have control over the direction of the line now called and the position of a point on
Now, however, instead of the position vector (red above) is given by Compute the object's speed, which is the magnitude of
In addition to the position vector it's often useful to display both the velocity and acceleration
These vectors, however, are usually drawn attached to the object itself, not the origin. The velocity vector always points in the direction of motion, but the same isn't true for acceleration.
Use the visualization and/or the derivative definitions to determine which of the two vectors (purple) and (green) is the acceleration for
Vectors also help us quantify nonlinear motion in the plane. Position vectors are formed by bundling and together into the ordered list velocity and acceleration vectors come from taking the first and second derivatives.
There's a great deal about vectors and motion we still have to learn but, to close out this quiz, let's start unraveling nonlinear motion in 2D.
Planet X orbits along a circle under the influence of gravity. If the circle has radius and is centered at the origin where the star sits, what's one possible position vector of the planet?
Hint: for all real
As Planet X orbits around its home star, it follows the position vector The visualization below shows three arrows representing the position of Planet X, its velocity, and its acceleration. Note that the red and blue vectors overlap; the blue one points towards the origin from the circle, while the red one points from the origin to the circle.
Which one corresponds to Planet X's acceleration
Now that we have some experience with 2D nonlinear motion, let's make the jump to 3D. Here's a visualization showing motion along a curve in 3D called a helix.
Here, measures the helix's vertical spacing. Taking gives us circular motion in the plane much like Planet X's. When the particle moves with constant speed parallel to the -axis while also simultaneously circulating around it.
Which vector represents the acceleration?
In the last quiz, we introduced vector fields where the number of components of the input matched that of the output. This quiz, however, showed how very versatile vectors truly are: here, we worked with vectors and depending only on a single variable
We'll explore the calculus of both kinds of vector functions (and others) in greater depth in Vector-valued Functions. There we'll apply our knowledge to save a magic harp from the clutches of a vile giant!
The next two quizzes go beyond motion and give a glimpse of how calculus can be synthesized together with the vector concept to produce new types of integrals and derivatives.