Abstract
Newton’s second law is the main tool you need to solve problems in mechanics. However, so far, we have only addressed motion and forces in one dimension. Fortunately, the description of motion and Newton’s laws do not change when we go to higher dimensions. We can continue to apply the structured problem-solving approach in exactly the same way as we did before. But in order to address motion in two- and three dimensions, we need to extend our mathematical and numerical methods to address two- and three-dimensional motion. This will be done in two parts: In this chapter we introduce general two- and three-dimensional motion. Later we will introduce constrained motion—motion constrained to occur along a specific path in the same way a rollercoaster cart is constrained to follow the rollercoaster track, or in the way a part of a rotating body is following the rotation of the body.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Our definition of a vector is rather limited compared to the more general definition of vector spaces you may be used to in mathematics. It means we usually limit ourselves to vectors in one, two, and three Cartesian dimensions. Usually, we illustrate a vector by an arrow, as shown in Fig. 6.1a. The length of the arrow indicates the magnitude, and the direction shows the direction of the vector. Notice that it does not matter where we start drawing a vector. The vector \(\mathbf {a}\) in Fig. 6.1 is the same even if it is drawn in position A, B, or C, but the vector \(\mathbf {b}\) in position D is different, because it has a different direction than \(\mathbf {a}\), and the vector \(\mathbf {c}\) in position E is different from both \(\mathbf {a}\) and \(\mathbf {b}\) since it has a different magnitude. Remember: The only thing that matters for a vector is its magnitude and direction—not where it starts.
- 2.
- 3.
Here we have implicitly assumed that the time derivatives of the unit vectors are zero. This is not necessarily the case: The unit vectors vary with time for rotating reference systems.
- 4.
- 5.
- 6.
- 7.
- 8.
- 9.
- 10.
- 11.
- 12.
- 13.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Malthe-Sorenssen, A. (2015). Motion in Two and Three Dimensions. In: Elementary Mechanics Using Matlab. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-19587-2_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-19587-2_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-19586-5
Online ISBN: 978-3-319-19587-2
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)